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A Troubled Time<br><br>
In mathematics a '''Steinberg symbol''' is a pairing function which generalises the [[Hilbert symbol]] and plays a role in the [[algebraic K-theory]] of [[field (mathematics)|fields]].  It is named after mathematician [[Robert Steinberg]].


From 1955 to 1965 there was a war right in the middle of America.  No, it wasn't a war like World War II or the Revolutionary War.  It was a war for the heart and soul of this country to determine once and for all if America was really going to be a land of equal opportunity for all.  It is a war that eventually took on the name of "The Civil Rights Movement."<br><br>We must make no mistake, this was not just a shouting match.  Some of the events that we even remember today became quite brutal and deadly.  Those who fought in this war on both sides were deadly serious about the causes they represented and willing to fight and even die to see their cause succeed.  The war waged for years and steady progress was made but not without tremendous sacrifice by the leaders of the movement who were committed to a giving a new meaning to the phrase "set my people free."<br><br>In all of black history, there may be no more significant a time since the Civil War when the rights of African Americans were so deeply fought and won. The tensions in the country had been building.  When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set.  But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America.  That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King.<br><br>This tremendous struggle for freedom was never easy and was often marked with violence.  Over the next ten years some of the most important milestone in black history took place including...<br><br>*    1957 - President Eisenhower had to send federal troops to Arkansas to secure admission to Central High School by nine black students.<br><br>*    1960 - The sit-in at Woolworths lunch counter in Greensboro North Carolina set the stage for nonviolent protest that was used with great success for the rest of the struggle.  Nonviolent protest and civil disobedience became a staple of the civil rights movement because of the influence of Martin Luther King.<br><br>*    1963 - The historic March on Washington in which over 200,000 people gathered to hear Dr. Kings famous "I Have a Dream" speech.<br><br>*    1964 - President Lyndon Johnson signed the bill that was the most significant event of his presidency and one he believed deeply in, the Civil Rights Act of 1964.<br><br>*   1965 - The assignation of Malcolm X and the Watts race rights.<br><br>*    1965 - President Johnson takes another bold step to accelerate the civil rights movement implementing Affirmative Action when he issues Executive Order 11246.<br><br>This short list is just a few of the highlights of this troubled time in which the rights of all citizens of American, black and white and of all colors were being redefined both on the streets, in the courts and in the different branches of government. In the years to come there would be great steps forward.  One by one, every area of American life would see breakthroughs by African Americans in the areas of sports, entertainment, education and politics.  There were many proud moments and there were moments of tremendous shame and heinous acts committed by both white and black people.  But through all that struggle, the society continued to grow and adapt to the will of the people as has always been the tradition in American culture.<br><br>The struggle is far from over.  Discrimination and hate speech continue to be a problem to this day.  And while it is easy to reflect on those days of struggle with regret, we can also look at them with pride.  We can be proud of the great leaders who demonstrated tremendous courage and wisdom to lead this nation to a better way of life.  And we can be proud of America because it is here where such a struggle can result in equality and freedom for all citizens, not just a few.<br><br>PPPPP 716A Troubled Time<br><br>From 1955 to 1965 there was a war right in the middle of America.  No, it wasn't a war like World War II or the Revolutionary War.  It was a war for the heart and soul of this country to determine once and for all if America was really going to be a land of equal opportunity for all.  It is a war that eventually took on the name of "The Civil Rights Movement."<br><br>We must make no mistake, this was not just a shouting match.  Some of the events that we even remember today became quite brutal and deadly.  Those who fought in this war on both sides were deadly serious about the causes they represented and willing to fight and even die to see their cause succeed.  The war waged for years and steady progress was made but not without tremendous sacrifice by the leaders of the movement who were committed to a giving a new meaning to the phrase "set my people free."<br><br>In all of black history, there may be no more significant a time since the Civil War when the rights of African Americans were so deeply fought and won.  The tensions in the country had been building.  When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set. But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America. That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King.<br><br>This tremendous struggle for freedom was never easy and was often marked with violence. Over the next ten years some of the most important milestone in black history took place including...<br><br>*   1957 - President Eisenhower had to send federal troops to Arkansas to secure admission to Central High School by nine black students.<br><br>*   1960 - The sit-in at Woolworths lunch counter in Greensboro North Carolina set the stage for nonviolent protest that was used with great success for the rest of the struggle.  Nonviolent protest and civil disobedience became a staple of the civil rights movement because of the influence of Martin Luther King.<br><br>*    1963 - The historic March on Washington in which over 200,000 people gathered to hear Dr. Kings famous "I Have a Dream" speech.<br><br>*    1964 - President Lyndon Johnson signed the bill that was the most significant event of his presidency and one he believed deeply in, the Civil Rights Act of 1964.<br><br>*    1965 - The assignation of Malcolm X and the Watts race rights.<br><br>*    1965 - President Johnson takes another bold step to accelerate the civil rights movement implementing Affirmative Action when he issues Executive Order 11246.<br><br>This short list is just a few of the highlights of this troubled time in which the rights of all citizens of American, black and white and of all colors were being redefined both on the streets, in the courts and in the different branches of government.  In the years to come there would be great steps forward.  One by one, every area of American life would see [http://Search.Huffingtonpost.com/search?q=breakthroughs&s_it=header_form_v1 breakthroughs] by African Americans in the areas of sports, entertainment, education and politics.  There were many proud moments and there were moments of tremendous shame and heinous acts committed by both white and black people.  But through all that struggle, the society continued to grow and adapt to the will of the people as has always been the tradition in American culture.<br><br>The struggle is far from over.  Discrimination and hate speech continue to be a problem to this day.  And while it is easy to reflect on those days of struggle with regret, we can also look at them with pride.  We can be proud of the great leaders who demonstrated tremendous courage and wisdom to lead this nation to a better way of life.  And we can be proud of America because it is here where such a struggle can result in equality and freedom for all citizens, not just a few.<br><br>PPPPP 716A Troubled Time<br><br>From 1955 to 1965 there was a war right in the middle of America.  No, it wasn't a war like World War II or the Revolutionary War.  It was a war for the heart and soul of this country to determine once and for all if America was really going to be a land of equal opportunity for all.  It is a war that eventually took on the name of "The Civil Rights Movement."<br><br>We must make no mistake, this was not just a shouting match.  Some of the events that we even remember today became quite brutal and deadly.  Those who fought in this war on both sides were deadly serious about the causes they represented and willing to fight and even die to see their cause succeed.  The war waged for years and steady progress was made but not without tremendous sacrifice by the leaders of the movement who were committed to a giving a new meaning to the phrase "set my people free."<br><br>In all of black history, there may be no more significant a time since the Civil War when the rights of African Americans were so deeply fought and won.  The tensions in the country had been building.  When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set.  But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America.  That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King.<br><br>This tremendous struggle for freedom was never easy and was often marked with violence.  Over the next ten years some of the most important milestone in black history took place including...<br><br>*    1957 - President Eisenhower had to send federal troops to Arkansas to secure admission to Central High School by nine black students.<br><br>*    1960 - The sit-in at Woolworths lunch counter in Greensboro North Carolina set the stage for nonviolent protest that was used with great success for the rest of the struggle.  Nonviolent protest and civil disobedience became a staple of the civil rights movement because of the influence of Martin Luther King.<br><br>*    1963 - The historic March on Washington in which over 200,000 people gathered to hear Dr. Kings famous "I Have a Dream" speech.<br><br>*    1964 - President Lyndon Johnson signed the bill that was the most significant event of his presidency and one he believed deeply in, the Civil Rights Act of 1964.<br><br>*    1965 - The assignation of Malcolm X and the  [http://tinyurl.com/pch83be ugg boots] Watts race rights.<br><br>*    1965 - President Johnson takes another bold step to accelerate the civil rights movement implementing Affirmative Action when he issues Executive Order 11246.<br><br>This short list is just a few of the highlights of this troubled time in which the rights of all citizens of American, black and white and of all colors were being redefined both on the streets, in the courts and in the different branches of government.  In the years to come there would be great steps forward.  One by one, every area of American life would see breakthroughs by African Americans in the areas of sports, entertainment, education and politics.  There were many proud moments and there were moments of tremendous shame and heinous acts committed by both white and black people.  But through all that struggle, the society continued to grow and adapt to the will of the people as has always been the tradition in American culture.<br><br>The struggle is far from over.  Discrimination and hate speech continue to be a problem to this day.  And while it is easy to reflect on those days of struggle with regret, we can also look at them with pride.  We can be proud of the great leaders who demonstrated tremendous courage and wisdom to lead this nation to a better way of life.  And we can be proud of America because it is here where such a struggle can result in equality and freedom for all citizens, not just a few.<br>A Troubled Time<br><br>From 1955 to 1965 there was a war right in the middle of America.  No, it wasn't a war like World War II or the Revolutionary War.  It was a war for the heart and soul of this country to determine once and for all if America was really going to be a land of equal opportunity for all.  It is a war that eventually took on the name of "The Civil Rights Movement."<br><br>We must make no mistake, this was not just a shouting match.  Some of the events that we even remember today became quite brutal and deadly.  Those who fought in this war on both sides were deadly serious about the causes they represented and willing to fight and even die to see their cause succeed.  The war waged for years and steady progress was made but not without tremendous sacrifice by the leaders of the movement who were committed to a giving a new meaning to the phrase "set my people free."<br><br>In all of black history, there may be no more significant a time since the Civil War when the rights of African Americans were so deeply fought and won.  The tensions in the country had been building.  When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set.  But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America.  That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King.<br><br>This tremendous struggle for freedom was never easy and was often marked with violence.  Over the next ten years some of the most important milestone in black history took place including...<br><br>*    1957 - President Eisenhower had to send federal troops to Arkansas to secure admission to Central High School by nine black students.<br><br>*    1960 - The sit-in at Woolworths lunch counter in Greensboro North Carolina set the stage for nonviolent protest that was used with great success for the rest of the struggle.  Nonviolent protest and civil disobedience became a staple of the civil rights movement because of the influence of Martin Luther King.<br><br>*    1963 - The historic March on Washington in which over 200,000 people gathered to hear Dr. Kings famous "I Have a Dream" speech.<br><br>*    1964 - President Lyndon Johnson signed the bill that was the most significant event of his presidency and one he believed deeply in, the Civil Rights Act of 1964.<br><br>*    1965 - The assignation of Malcolm X and the Watts race rights.<br><br>*    1965 - President Johnson takes another bold step to accelerate the civil rights movement implementing Affirmative Action when he issues Executive Order 11246.<br><br>This short list is just a few of the highlights of this troubled time in which the rights of all citizens of American, black and white and of all colors were being redefined both on the streets, in the courts and in the different branches of government.  In the years to come there would be great steps forward.  One by one, every area of American life would see breakthroughs by African Americans in the areas of sports, entertainment, education and politics.  There were many proud moments and there were moments of tremendous shame and heinous acts committed by both white and black people.  But through all that struggle, the society continued to grow and adapt to the will of the people as has always been the tradition in American culture.<br><br>The struggle is far from over.  Discrimination and hate speech continue to be a problem to this day.  And while it is easy to reflect on those days of struggle with regret, we can also look at them with pride.  We can be proud of the great leaders who demonstrated tremendous courage and wisdom to lead this nation to a better way of life.  And we can be proud of America because it is here where such a struggle can result in equality and freedom for all citizens, not just a few.<br><br>PPPPP 716<br>A Troubled Time<br><br>From 1955 to 1965 there was a war right in the middle of America.  No, it wasn't a war like World War II or the Revolutionary War.  It was a war for the heart and soul of this country to determine once and for all if America was really going to be a land of equal opportunity for all.  It is a war that eventually took on the name of "The Civil Rights Movement."<br><br>We must make no mistake, this was not just a shouting match.  Some of the events that we even remember today became quite brutal and deadly.  Those who fought in this war on both sides were deadly serious about the causes they represented and willing to fight and even die to see their cause succeed.  The war waged for years and steady progress was made but not without tremendous sacrifice by the leaders of the movement who were committed to a giving a new meaning to the phrase "set my people free."<br><br>In all of black history, there may be no more significant a time since the Civil War when the rights of African Americans were so deeply fought and won.  The tensions in the country had been building.  When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set.  But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America.  That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King.<br><br>This tremendous struggle for freedom was never easy and was often marked with violence.  Over the next ten years some of the most important milestone in black history took place including...<br><br>*    1957 - President Eisenhower had to send federal troops to Arkansas to secure admission to Central High School by nine black students.<br><br>*    1960 - The sit-in at Woolworths lunch counter in Greensboro North Carolina set the stage for nonviolent protest that was used with great success for the rest of the struggle.  Nonviolent protest and civil disobedience became a staple of the civil rights movement because of the influence of Martin Luther King.<br><br>*    1963 - The historic March on Washington in which over 200,000 people gathered to hear Dr. Kings famous "I Have a Dream" speech.<br><br>*    1964 - President Lyndon Johnson signed the bill that was the most significant event of his presidency and one he believed deeply in, the Civil Rights Act of 1964.<br><br>*    1965 - The assignation of Malcolm X and the Watts race rights.<br><br>*    1965 - President Johnson takes another bold step to accelerate the civil rights movement implementing Affirmative Action when he issues Executive Order 11246.<br><br>This short list is just a few of the highlights of this troubled time in which the rights of all citizens of American, black and white and of all colors were being redefined both on the streets, in the courts and in the different branches of government.  In the years to come there would be great steps forward.  One by one, every area of American life would see breakthroughs by African Americans in the areas of sports, entertainment, education and politics.  There were many proud moments and there were moments of tremendous shame and heinous acts committed by both white and black people.  But through all that struggle, the society continued to grow and adapt to the will of the people as has always been the tradition in American culture.<br><br>The struggle is far from over.  Discrimination and hate speech continue to be a problem to this day.  And while it is easy to reflect on those days of struggle with regret, we can also look at them with pride.  We can be proud of the great leaders who demonstrated tremendous courage and wisdom to lead this nation to a better way of life.  And we can be proud of America because it is here where such a struggle can result in equality and freedom for all citizens, not just a few.<br><br>PPPPP 716<br>A Troubled Time<br><br>From 1955 to 1965 there was a war right in the middle of America.  No, it wasn't a war like World War II or the Revolutionary War.  It was a war for the heart and soul of this country to determine once and for all if America was really going to be a land of equal opportunity for all.  It is a war that eventually took on the name of "The Civil Rights Movement."<br><br>We must make no mistake, this was not just a shouting match.  Some of the events that we even remember today became quite brutal and deadly.  Those who fought in this war on both sides were deadly serious about the causes they represented and willing to fight and even die to see their cause succeed.  The war waged for years and steady progress was made but not without tremendous sacrifice by the leaders of the movement who were committed to a giving a new meaning to the phrase "set my people free."<br><br>In all of black history, there may be no more significant a time since the Civil War when the rights of African Americans were so deeply fought and won.  The tensions in the country had been building.  When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set.  But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America.  That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King.<br><br>This tremendous struggle for freedom was never easy and was often marked with violence.  Over the next ten years some of the most important milestone in black history took place including...<br><br>*    1957 - President Eisenhower had to send federal troops to Arkansas to secure admission to Central High School by nine black students.<br><br>*    1960 - The sit-in at Woolworths lunch counter in Greensboro North Carolina set the stage for nonviolent protest that was used with great success for the rest of the struggle.  Nonviolent protest and civil disobedience became a staple of the civil rights movement because of the influence of Martin Luther King.<br><br>*    1963 - The historic March on Washington in which over 200,000 people gathered to hear Dr. Kings famous "I Have a Dream" speech.<br><br>*    1964 - President Lyndon Johnson signed the bill that was the most significant event of his presidency and one he believed deeply in, the Civil Rights Act of 1964.<br><br>*    1965 - The assignation of Malcolm X and the Watts race rights.<br><br>*    1965 - President Johnson takes another bold step to accelerate the civil rights movement implementing Affirmative Action when he issues Executive Order 11246.<br><br>This short list is just a few of the highlights of this troubled time in which the rights of all citizens of American, black and white and of all colors were being redefined both on the streets, in the courts and in the different branches of government.  In the years to come there would be great steps forward.  One by one, every area of American life would see breakthroughs by African Americans in the areas of sports, entertainment, education and politics.  There were many proud moments and there were moments of tremendous shame and heinous acts committed by both white and black people.  But through all that struggle, the society continued to grow and adapt to the will of the people as has always been the tradition in American culture.<br><br>The struggle is far from over.  Discrimination and hate speech continue to be a problem to this day.  And while it is easy to reflect on those days of struggle with regret, we can also look at them with pride.  We can be proud of the great leaders who demonstrated tremendous courage and wisdom to lead this nation to a better way of life.  And we can be proud of America because it is here where such a struggle can result in equality and freedom for all citizens, not just a few.<br><br>PPPPP 716<br>A Troubled Time<br><br>From 1955 to 1965 there was a war right in the middle of America.  No, it wasn't a war like World War II or the Revolutionary War.  It was a war for the heart and soul of this country to determine once and for all if America was really going to be a land of equal opportunity for all.  It is a war that eventually took on the name of "The Civil Rights Movement."<br><br>We must make no mistake, this was not just a shouting match.  Some of the events that we even remember today became quite brutal and deadly.  Those who fought in this war on both sides were deadly serious about the causes they represented and willing to fight and even die to see their cause succeed.  The war waged for years and steady progress was made but not without tremendous sacrifice by the leaders of the movement who were committed to a giving a new meaning to the phrase "set my people free."<br><br>In all of black history, there may be no more significant a time since the Civil War when the rights of African Americans were so deeply fought and won.  The tensions in the country had been building.  When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set.  But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America.  That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King.<br><br>This tremendous struggle for freedom was never easy and was often marked with violence.  Over the next ten years some of the most important milestone in black history took place including...<br><br>*    1957 - President Eisenhower had to send federal troops to Arkansas to secure admission to Central High School by nine black students.<br><br>*    1960 - The sit-in at Woolworths lunch counter in Greensboro North Carolina set the stage for nonviolent protest that was used with great success for the rest of the struggle.  Nonviolent protest and civil disobedience became a staple of the civil rights movement because of the influence of Martin Luther King.<br><br>*    1963 - The historic March on Washington in which over 200,000 people gathered to hear Dr. Kings famous "I Have a Dream" speech.<br><br>*    1964 - President Lyndon Johnson signed the bill that was the most significant event of his presidency and one he believed deeply in, the Civil Rights Act of 1964.<br><br>*    1965 - The assignation of Malcolm X and the Watts race rights.<br><br>*    1965 - President Johnson takes another bold step to accelerate the civil rights movement implementing Affirmative Action when he issues Executive Order 11246.<br><br>This short list is just a few of the highlights of this troubled time in which the rights of all citizens of American, black and white and of all colors were being redefined both on the streets, in the courts and in the different branches of government.  In the years to come there would be great steps forward.  One by one, every area of American life would see breakthroughs by African Americans in the areas of sports, entertainment, education and politics.  There were many proud moments and there were moments of tremendous shame and heinous acts committed by both white and black people.  But through all that struggle, the society continued to grow and adapt to the will of the people as has always been the tradition in American culture.<br><br>The struggle is far from over.  Discrimination and hate speech continue to be a problem to this day.  And while it is easy to reflect on those days of struggle with regret, we can also look at them with pride.  We can be proud of the great leaders who demonstrated tremendous courage and wisdom to lead this nation to a better way of life.  And we can be proud of America because it is here where such a struggle can result in equality and freedom for all citizens, not just a few.<br><br>PPPPP 716<br>A Troubled Time<br><br>From 1955 to 1965 there was a war right in the middle of America.  No, it wasn't a war like World War II or the Revolutionary War.  It was a war for the heart and soul of this country to determine once and for all if America was really going to be a land of equal opportunity for all.  It is a war that eventually took on the name of "The Civil Rights Movement."<br><br>We must make no mistake, this was not just a shouting match.  Some of the events that we even remember today became quite brutal and deadly.  Those who fought in this war on both sides were deadly serious about the causes they represented and willing to fight and even die to see their cause succeed.  The war waged for years and steady progress was made but not without tremendous sacrifice by the leaders of the movement who were committed to a giving a new meaning to the phrase "set my people free."<br><br>In all of black history, there may be no more significant a time since the Civil War when the rights of African Americans were so deeply fought and won.  The tensions in the country had been building.  When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set.  But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America.  That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King.<br><br>This tremendous struggle for freedom was never easy and was often marked with violence.  Over the next ten years some of the most important milestone in black history took place including...<br><br>*    1957 - President Eisenhower had to send federal troops to Arkansas to secure admission to Central High School by nine black students.<br><br>*    1960 - The sit-in at Woolworths lunch counter in Greensboro North Carolina set the stage for nonviolent protest that was used with great success for the rest of the struggle.  Nonviolent protest and civil disobedience became a staple of the civil rights movement because of the influence of Martin Luther King.<br><br>*    1963 - The historic March on Washington in which over 200,000 people gathered to hear Dr. Kings famous "I Have a Dream" speech.<br><br>*    1964 - President Lyndon Johnson signed the bill that was the most significant event of his presidency and one he believed deeply in, the Civil Rights Act of 1964.<br><br>*    1965 - The assignation of Malcolm X and the Watts race rights.<br><br>*    1965 - President Johnson takes another bold step to accelerate the civil rights movement implementing Affirmative Action when he issues Executive Order 11246.<br><br>This short list is just a few of the highlights of this troubled time in which the rights of  [http://tinyurl.com/pch83be ugg boots] all citizens of American, black and white and of all colors were being redefined both on the streets, in the courts and in the different branches of government.  In the years to come there would be great steps forward.  One by one, every area of American life would see breakthroughs by African Americans in the areas of sports, entertainment, education and politics.  There were many proud moments and there were moments of tremendous shame and heinous acts committed by both white and black people.  But through all that struggle, the society continued to grow and adapt to the will of the people as has always been the tradition in American culture.<br><br>The struggle is far from over.  Discrimination and hate speech continue to be a problem to this day.  And while it is easy to reflect on those days of struggle with regret, we can also look at them with pride.  We can be proud of the great leaders who demonstrated tremendous courage and wisdom to lead this nation to a better way of life.  And we can be proud of America because it is here where such a struggle can result in equality and freedom for all citizens, not just a few.<br><br>PPPPP 716<br>A Troubled Time<br><br>From 1955 to 1965 there was a war right in the middle of America.  No, it wasn't a war like World War II or the Revolutionary War.  It was a war for the heart and soul of this country to determine once and for all if America was really going to be a land of equal opportunity for all.  It is a war that eventually took on the name of "The Civil Rights Movement."<br><br>We must make no mistake, this was not just a shouting match.  Some of the events that we even remember today became quite brutal and deadly.  Those who fought in this war on both sides were deadly serious about the causes they represented and willing to fight and even die to see their cause succeed.  The war waged for years and steady progress was made but not without tremendous sacrifice by the leaders of the movement who were committed to a giving a new meaning to the phrase "set my people free."<br><br>In all of black history, there may be no more significant a time since the Civil War when the rights of African Americans were so deeply fought and won.  The tensions in the country had been building.  When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set.  But it was on December 1, 1955 when Rosa  [http://tinyurl.com/pch83be http://tinyurl.com/pch83be] Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America.  That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King.<br><br>This tremendous struggle for freedom was never easy and was often marked with violence.  Over the next ten years some of the most important milestone in black history took place including...<br><br>*    1957 - 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President Eisenhower had to send federal troops to Arkansas to secure admission to Central High School by nine black students.<br><br>*    1960 - The sit-in at Woolworths lunch counter in Greensboro North Carolina set the stage for nonviolent protest that was used with great success for the rest of the struggle.  Nonviolent protest and civil disobedience became a staple of the civil rights movement because of the influence of Martin Luther King.<br><br>*    1963 - The historic March on Washington in which over 200,000 people gathered to hear Dr. Kings famous "I Have a Dream" speech.<br><br>*    1964 - President Lyndon Johnson signed the bill that was the most significant event of his presidency and one he believed deeply in, the Civil Rights Act of 1964.<br><br>*    1965 - The assignation of Malcolm X and the Watts race rights.<br><br>*    1965 - President Johnson takes another bold step to accelerate the civil rights movement implementing Affirmative Action when he issues Executive Order 11246.<br><br>This short list is just a few of the highlights of this troubled time in which the rights of all citizens of American, black and white and of all colors were being redefined both on the streets, in the courts and in the different branches of government.  In the years to come there would be great steps forward.  One by one, every area of American life would see breakthroughs by African Americans in the areas of sports, entertainment, education and politics.  There were many proud moments and there were moments of tremendous shame and heinous acts committed by  [http://tinyurl.com/pch83be http://tinyurl.com/pch83be] both white and black people.  But through all that struggle, the society continued to grow and adapt to the will of the people as has always been the tradition in American culture.<br><br>The struggle is far from over.  Discrimination and hate speech continue to be a problem to this day.  And while it is easy to reflect on those days of struggle with regret, we can also look at them with pride.  We can be proud of the great leaders who demonstrated tremendous courage and wisdom to lead this nation to a better way of life.  And we can be proud of America because it is here where such a struggle can result in equality and freedom for all citizens, not just a few.<br><br>PPPPP 716<br>arvinoor<br>arvinoor siregar<br>arvinoor siregar sh.<br>arvinoor siregar sh.mh.<br>PPPPP 716
For a field ''F'' we define a ''Steinberg symbol'' (or simply a ''symbol'') to be a function
<math>( \cdot , \cdot ) : F^* \times F^* \rightarrow G</math>, where ''G'' is an abelian group, written multiplicatively, such that
* <math>( \cdot , \cdot ) </math> is bimultiplicative;
* if <math>a+b = 1</math> then <math>(a,b) = 1</math>.
 
The symbols on ''F'' derive from a "universal" symbol, which may be regarded as taking values in <math>F^* \otimes F^* / \langle a \otimes 1-a \rangle</math>.  By a theorem of Matsumoto, this group is <math>K_2 F</math> and is part of the [[Milnor K-theory]] for a field.
 
==Properties==
If (⋅,⋅) is a symbol then (assuming all terms are defined)
* <math> (a, -a) = 1 </math>;
* <math> (b, a) = (a, b)^{-1} </math>;
* <math> (a, a) = (a, -1) </math> is an element of order 1 or 2;
* <math> (a, b) = (a+b, -b/a) </math>.
 
==Examples==
* The trivial symbol which is identically 1.
 
* The [[Hilbert symbol]] on ''F'' with values in {±1} defined by<ref>{{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=A Course in Arithmetic | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | volume=7 | isbn=978-3-540-90040-5 | year=1996 }}</ref><ref name=Mil94>Milnor (1971) p.94</ref>
:<math>(a,b)=\begin{cases}1,&\mbox{ if }z^2=ax^2+by^2\mbox{ has a non-zero solution }(x,y,z)\in F^3;\\-1,&\mbox{ if  not.}\end{cases}</math>
 
*The [[Contou-Carrère symbol]] is a symbol for the ring of [[Laurent power series]] over an [[Artinian ring]].
 
==Continuous symbols==
If ''F'' is a [[topological field]] then a symbol ''c'' is ''weakly continuous'' if for each ''y'' in ''F''<sup>&lowast;</sup> the set of ''x'' in ''F''<sup>&lowast;</sup> such that ''c''(''x'',''y'') = 1 is [[closed set|closed]] in ''F''<sup>&lowast;</sup>.  This makes no reference to a topology on the codomain ''G''. If ''G'' is a [[topological group]], then one may speak of a ''continuous symbol'', and when ''G'' is [[Hausdorff space|Hausdorff]] then a continuous symbol is weakly continuous.<ref name=Mil165>Milnor (1971) p.165</ref>
 
The only weakly continuous symbols on '''R''' are the trivial symbol and the Hilbert symbol: the only weakly continuous symbol on '''C''' is the trivial symbol.<ref name=Mil166>Milnor (1971) p.166</ref> The characterisation of weakly continuous symbols on a non-Archimedean [[local field]] ''F'' was obtained by Moore.  The group K<sub>2</sub>(''F'') is the direct sum of a [[cyclic group]] of order ''m'' and a [[divisible group]] K<sub>2</sub>(''F'')<sup>''m''</sup>.  A symbol on ''F'' lifts to a homomorphism on K<sub>2</sub>(''F'') and is weakly continuous precisely when it annihilates the divisible component K<sub>2</sub>(''F'')<sup>''m''</sup>.  It follows that every weakly continuous symbol factors through the [[norm residue symbol]].<ref name=Mil175>Milnor (1971) p.175</ref>
 
==See also==
* [[Steinberg group (K-theory)]]
 
==References==
{{reflist}}
* {{cite book | first1=P.E. | last1=Conner | first2=R. | last2=Perlis | title=A Survey of Trace Forms of Algebraic Number Fields | series=Series in Pure Mathematics | volume=2 | publisher=World Scientific | year=1984 | isbn=9971-966-05-0 | zbl=0551.10017 }}
* {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | authorlink=Tsit Yuen Lam | publisher=American Mathematical Society | year=2005 | isbn=0-8218-1095-2 | pages=132–142 | zbl=1068.11023 }}
* {{cite book | last1=Milnor | first1=John Willard | author1-link= John Milnor | title=Introduction to algebraic K-theory | publisher=[[Princeton University Press]] | location=Princeton, NJ | mr=0349811 | year=1971 | zbl=0237.18005 | series=Annals of Mathematics Studies | volume=72 }}
* {{cite journal | first=Robert | last=Steinberg | authorlink=Robert Steinberg | title=Générateurs, relations et revêtements de groupes algébriques | journal=Colloq. Théorie des Groupes Algébriques | location=Bruxelles | year=1962 | publisher=Gauthier-Villars | pages=113–127 | mr=MR0153677 | zbl=0272.20036 | language=French }}
 
==External links==
*[http://eom.springer.de/S/s130540.htm Steinberg symbol] at the [[Encyclopaedia of Mathematics]]
 
[[Category:K-theory]]

Revision as of 12:57, 9 April 2013

In mathematics a Steinberg symbol is a pairing function which generalises the Hilbert symbol and plays a role in the algebraic K-theory of fields. It is named after mathematician Robert Steinberg.

For a field F we define a Steinberg symbol (or simply a symbol) to be a function , where G is an abelian group, written multiplicatively, such that

The symbols on F derive from a "universal" symbol, which may be regarded as taking values in . By a theorem of Matsumoto, this group is and is part of the Milnor K-theory for a field.

Properties

If (⋅,⋅) is a symbol then (assuming all terms are defined)

Examples

  • The trivial symbol which is identically 1.

Continuous symbols

If F is a topological field then a symbol c is weakly continuous if for each y in F∗ the set of x in F∗ such that c(x,y) = 1 is closed in F∗. This makes no reference to a topology on the codomain G. If G is a topological group, then one may speak of a continuous symbol, and when G is Hausdorff then a continuous symbol is weakly continuous.[3]

The only weakly continuous symbols on R are the trivial symbol and the Hilbert symbol: the only weakly continuous symbol on C is the trivial symbol.[4] The characterisation of weakly continuous symbols on a non-Archimedean local field F was obtained by Moore. The group K2(F) is the direct sum of a cyclic group of order m and a divisible group K2(F)m. A symbol on F lifts to a homomorphism on K2(F) and is weakly continuous precisely when it annihilates the divisible component K2(F)m. It follows that every weakly continuous symbol factors through the norm residue symbol.[5]

See also

References

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External links

  1. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  2. ↑ Milnor (1971) p.94
  3. ↑ Milnor (1971) p.165
  4. ↑ Milnor (1971) p.166
  5. ↑ Milnor (1971) p.175
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