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The Artin–Schreier theorem for extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110740/a1107401.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110740/a1107402.png" /> of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110740/a1107403.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110740/a1107404.png" /> states that every such Galois extension is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110740/a1107405.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110740/a1107406.png" /> is the root of a polynomial of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110740/a1107407.png" />, an Artin–Schreier polynomial.
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{{TEX|done}}{{MSC|12E10}}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110740/a1107408.png" /> is a function field, these polynomials define Artin–Schreier curves, which in turn give rise to Artin–Schreier codes (cf. [[Artin–Schreier code|Artin–Schreier code]]).
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The Artin–Schreier theorem for extensions $K$ of degree $p$ of a field $F$ of characteristic $p>0$ states that every such Galois extension is of the form $K = F(\alpha)$, where $\alpha$ is the root of a polynomial of the form $X^p - X - a$, an Artin–Schreier polynomial.
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The function $A : X \mapsto X^p - X$ is $p$-to-one  since $A(x) = A(x+1)$.  It is in fact $\mathbf{F}_p$-linear on $F$ as a  [[vector space]], with kernel the one-dimensional subspace generated by  $1_F$, that is, $\mathbf{F}_p$ itself.
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Suppose that  $F$ is finite of characteristic $p$.  The [[Frobenius map]] is an  [[field automorphism|automorphism]] of $F$ and so its [[inverse  function|inverse]], the $p$-th root map, is defined everywhere, and  $p$-th roots do not generate any non-trivial extensions.
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If  $F$ is finite, then $A$ is exactly $p$-to-1 and the image of $A$ is a  $\mathbf{F}_p$-subspace of codimension 1. There is always some element  $a \in F$ not in the image of $A$, and so the corresponding  Artin-Schreier polynomial has no root in $F$: it is an [[irreducible  polynomial]] and the [[quotient ring]] $F[X]/\langle A_\alpha(X)  \rangle$ is a field which is a degree $p$ extension of $F$. Since  finite fields of the same order are unique up to isomorphism, we may say  that this is "the" degree $p$ extension of $F$.  As before, both roots  of the equation lie in the extension, which is thus a ''[[splitting  field]]'' for the equation and hence a [[Galois extension]]: in this  case the roots are of the form $\beta,\,\beta+1, \ldots,\beta+(p-1)$.
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If $F$ is a function field, these polynomials define Artin–Schreier curves, which in turn give rise to Artin–Schreier codes (cf. [[Artin–Schreier code|Artin–Schreier code]]).
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====References====
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{|
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|valign="top"|{{Ref|La}}|| S. Lang,  "Algebra" , Addison-Wesley  (1974)
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====Comment====
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This is also a name for the theorem that a field is [[Formally real field|formally real]] (can be [[ordered field|ordered]]) if and only if $-1$ is not a sum of squares.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Lang,   "Algebra" , Addison-Wesley  (1974)</TD></TR></table>
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* J.W. Milnor, D. Husemöller, ''Symmetric bilinear forms'', Ergebnisse der Mathematik und ihrer Grenzgebiete '''73''', Springer-Verlag (1973) p.60 ISBN 0-387-06009-X {{ZBL|0292.10016}}

Revision as of 20:05, 25 November 2016

2020 Mathematics Subject Classification: Primary: 12E10 [MSN][ZBL]

The Artin–Schreier theorem for extensions $K$ of degree $p$ of a field $F$ of characteristic $p>0$ states that every such Galois extension is of the form $K = F(\alpha)$, where $\alpha$ is the root of a polynomial of the form $X^p - X - a$, an Artin–Schreier polynomial.

The function $A : X \mapsto X^p - X$ is $p$-to-one since $A(x) = A(x+1)$. It is in fact $\mathbf{F}_p$-linear on $F$ as a vector space, with kernel the one-dimensional subspace generated by $1_F$, that is, $\mathbf{F}_p$ itself.

Suppose that $F$ is finite of characteristic $p$. The Frobenius map is an automorphism of $F$ and so its inverse, the $p$-th root map, is defined everywhere, and $p$-th roots do not generate any non-trivial extensions.

If $F$ is finite, then $A$ is exactly $p$-to-1 and the image of $A$ is a $\mathbf{F}_p$-subspace of codimension 1. There is always some element $a \in F$ not in the image of $A$, and so the corresponding Artin-Schreier polynomial has no root in $F$: it is an irreducible polynomial and the quotient ring $F[X]/\langle A_\alpha(X) \rangle$ is a field which is a degree $p$ extension of $F$. Since finite fields of the same order are unique up to isomorphism, we may say that this is "the" degree $p$ extension of $F$. As before, both roots of the equation lie in the extension, which is thus a splitting field for the equation and hence a Galois extension: in this case the roots are of the form $\beta,\,\beta+1, \ldots,\beta+(p-1)$.

If $F$ is a function field, these polynomials define Artin–Schreier curves, which in turn give rise to Artin–Schreier codes (cf. Artin–Schreier code).

References

[La] S. Lang, "Algebra" , Addison-Wesley (1974)

Comment

This is also a name for the theorem that a field is formally real (can be ordered) if and only if $-1$ is not a sum of squares.

References

  • J.W. Milnor, D. Husemöller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete 73, Springer-Verlag (1973) p.60 ISBN 0-387-06009-X Zbl 0292.10016
How to Cite This Entry:
Artin-Schreier theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Artin-Schreier_theorem&oldid=17088
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article