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| :''See [[Artin–Schreier theorem]] for theory about [[real-closed field]]s.''
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| In [[mathematics]], '''Artin–Schreier theory''' is a branch of [[Galois theory]], and more specifically is a positive [[characteristic (algebra)|characteristic]] analogue of [[Kummer theory]], for Galois [[Field extension|extensions]] of degree equal to the characteristic ''p''. {{harvs|txt|author1-link=Emil Artin|author2-link=Otto Schreier|last1=Artin|last2=Schreier|year=1927}} introduced Artin–Schreier theory for extensions of prime degree ''p'', and {{harvs|txt|authorlink=Ernst Witt|last=Witt|year=1936}} generalized it to extensions of prime power degree ''p''<sup>''n''</sup>.
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| If ''K'' is a [[field (mathematics)|field]] of characteristic ''p'', a [[prime number]], any [[polynomial]] of the form
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| :<math>X^p - X + \alpha,\,</math>
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| for <math>\alpha</math> in ''K'', is called an ''Artin–Schreier polynomial''. When <math>\alpha</math> does not lie in the subset <math> \{ y \in K \, | \, y=x^p-x \; \mbox{for } x \in K \}</math>, this polynomial is [[irreducible]] in ''K''[''X''], and its [[splitting field]] over ''K'' is a [[cyclic extension]] of ''K'' of degree ''p''. This follows since for any root β, the numbers β + i, for <math>1\le i\le p</math>, form all the roots—by [[Fermat's little theorem]]—so the splitting field is <math> K(\beta) </math>.
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| Conversely, any Galois extension of ''K'' of degree ''p'' equal to the characteristic of ''K'' is the splitting field of an Artin–Schreier polynomial. This can be proved using additive counterparts of the methods involved in [[Kummer theory]], such as [[Hilbert's theorem 90]] and additive [[Galois cohomology]]. These extensions are called ''Artin–Schreier extensions''.
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| Artin–Schreier extensions play a role in the theory of [[solvability by radicals]], in characteristic ''p'', representing one of the possible classes of extensions in a solvable chain.
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| They also play a part in the theory of [[abelian varieties]] and their [[isogeny|isogenies]]. In characteristic ''p'', an isogeny of degree ''p'' of abelian varieties must, for their function fields, give either an Artin–Schreier extension or a [[purely inseparable extension]].
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| ==Artin–Schreier–Witt extensions==
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| There is an analogue of Artin–Schreier theory which describes cyclic extensions in characteristic ''p'' of ''p''-power degree (not just degree ''p'' itself), using
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| [[Witt vector]]s, developed by {{harvs|txt|authorlink=Ernst Witt|last=Witt|year=1936}}.
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| ==References==
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| *{{Citation | last1=Artin | first1=Emil | author1-link=Emil Artin | last2=Schreier | first2=Otto | author2-link=Otto Schreier | title=Eine Kennzeichnung der reell abgeschlossenen Körper | publisher=Springer Berlin / Heidelberg | year=1927 | journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | issn=0025-5858 | volume=5 | pages=225–231 | doi=10.1007/BF02952522}}
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| * {{Lang Algebra|edition=3r}} Section VI.6
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| * {{Neukirch et al. CNF}} Section VI.1
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| * {{Citation | url=http://www.digizeitschriften.de/main/dms/img/?IDDOC=504725 | last1=Witt | first1=Ernst | author1-link = Ernst Witt | title=Zyklische Körper und Algebren der Characteristik p vom Grad p<sup>n</sup>. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik p<sup>n</sup> | language=German | year=1936 | journal=[[Journal für die reine und angewandte Mathematik]] | volume=176 | pages=126–140}}
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| {{DEFAULTSORT:Artin-Schreier theory}}
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| [[Category:Galois theory]]
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