The following is claimed in the proof of Theorem 7.5 of Auslander, Goldman, "The Brauer group of a commutative ring":



> 
> Let $k$ be a nonperfect field of positive characteristic $p$, let $K := k(x)$ be the function field in one variable, and let $L := K[y]/(y^{p}-y-x)$ be the Artin-Schreier extension of $K$ associated to $x \in K$. For any $c \in k^{\times} \setminus (k^{\times})^{p}$, there does not exist $\ell \in L$ such that $c = \mathrm{Nm}\_{L/K}(\ell)$.
> 
> 
> 


Why is this true?


*Remarks*: 


1. An arbitrary element of $L$ is of the form $\ell = f(y)$ for some polynomial $$ f(T) = a\_{0} + a\_{1}T + \dotsb + a\_{p-1}T^{p-1} $$ of degree at most $p-1$ with $a\_{i} \in K$, and in this case we have norm $\mathrm{Nm}\_{L/K}(\ell) = \prod\_{i=0}^{p-1} f(y+i)$. However I am not sure if this product simplifies to a nice expression in $x$ and the $a\_{i}$. For $p=2$ the product is $(a\_{0}^{2} + xa\_{1}^{2}) + a\_{0}a\_{1}$ and for $p=3$ the product is $(a\_{0}^{3} + xa\_{1}^{3} + x^{2}a\_{2}^{3}) + (-a\_{0}a\_{1}^{2} - a\_{0}^{2}a\_{2} + a\_{0}a\_{2}^{2} - xa\_{1}a\_{2}^{2})$.
2. See also Example 11.6.9 of Fried, Jarden, "Field Arithmetic" which treats the $p=2$ case.


*Keywords*: Artin-Schreier extensions, Galois, cyclic, norm



Norms of elements in Artin-Schreier extensions
Let's define the leading coefficient of a rational function to be the leading coefficient of the numerator over the leading coefficient of the denominator. We will show that the leading term of any norm is a $p$th power. It will follow that any norm that is a constant is a $p$th power.


Observe that $L$ is isomorphic to $k(y)$ . For $l(y)$ a rational function in $y$, the norm of $l$ is $l(y)l(y+1) \dots l(y+p-1)$, which are all rational functions with the same leading term, so the leading term of their product is the $p$th power of the leading term of $y$. Then we can write it as a rational function in $x$ by the substitution $y^p-y=x$, which does not affect the leading term, so the leading term is a $p$th power, as desired.


