# Artin-Schreier theorem

2010 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%E2%80%93Schreier_theorem&oldid=22030