Difference between revisions of "Artin-Schreier theorem"
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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. | 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. | 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 [[field automorphism|automorphism]] of $ | + | 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. |
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 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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====References==== | ====References==== | ||
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| − | + | |valign="top"|{{Ref|La}}|| S. Lang, "Algebra" , Addison-Wesley (1974) | |
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| + | |} | ||
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| + | ====Comment==== | ||
| + | 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. | ||
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| + | ====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}} | ||
Latest revision as of 19:41, 15 November 2023
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
Artin-Schreier theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Artin-Schreier_theorem&oldid=30362