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I have a question from Serre's book "Corps Locaux", namely question 5a in section 2 of chapter IV. It is as follows:

"Let $e_K$ be the absolute ramification index of K, and let n be a positive integer prime to p and (strictly) less than $pe_K/(p-1)$; let y be an element of valuation -n. Show that the Artin-Schreier equation $x^p - x =y$ is irreducible over K, and defines an extension L/K which is cyclic of degree p. (Show that if x is a root of this equation, then the other roots have the form $x + z_i$ $(0 \leq i < p$, with $z_i \in A_K$, and $z_i \equiv i \mod{\mathbb{p_L}}$."

Here $K$ is characteristic 0, its residue field has positive characteristic p, and $K$ is a complete local field.

I'm struggling with the last part, namely showing the roots have the given form. Right now, as $(n,p)=1$, if $x$ is any root then $val(x^p) = -n$, which forces any root to not be within $K$. If I can show that in the extension of $K$ generated by $x$ I can write all other roots in the desired form, this will mean $K(x)$ is the splitting field and the Galois group will act transitively on the roots, whence the result.

Could I be pointed in the right direction?

Thanks,

Garnet

Garnet
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1 Answers1

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We will show that the other roots are of the form $x + z_i$ for $z_i \in A_L$, the integral closure of $A_K$ in $A_L$. To do this, we just need to show the $z_i$ are roots of a monic equation with coefficients in $A_L$, since $A_L$ is integrally closed. Our only condition is $x^p - x - y = 0$, so we take the difference of $x^p-x-y$ and $(x+z)^p - (x+z) - y$ to obtain $$ 0 = (x+z)^p - x^p - (x+z) + x = -z + \sum_{i=1}^p \left(\binom{p}{i} x^{p-i}\right) z^i.$$ Now use your conditions to show that this equation is what we desire.

References: "Local Fields and Their Extensions" by I.B. Fesenko and S.V. Vostokov, III.2.5

Joshua Mundinger
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