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# Help wanted with CebotarevChebotarev condition in characteristic 2

Having promised a longtime collaborator that I would clear my plate to finish up some joint work of ours, I am swallowing my pride and tossing up the following technical point of function field arithmetic to the experts:

Let $k_0 = \mathbb{F}_2(t)$, and let $k(P)$ be a finite separable extension of $k_0$. Let $S$ be a finite set of places $v$ of $k_0$ (thus corresponding to monic irreducible polynomials in $\mathbb{F}_2[t]$ together with possibly the place at infinity). Consider the set of monic irreducible polynomials $w \in \mathbb{F}_2[t]$ with the following properties:

(i) Every place $v$ in $S$ splits completely in the quadratic Artin-Schreier extension of $k_0$ defined by the polynomial $X^2 + X = \frac{1}{w}$, and
(ii) $w$ splits completely in $k(P)$.

Problem: Show that there are infinitely many such $w$.

The natural strategy is to show that the set of such $w$ satisfies some Cebotarev Chebotarev condition -- presumably it even contains the set of all primes splitting in a certain finite Galois extension of $k_0$ -- and then apply the Cebotarev Chebotarev Density Theorem in this context (e.g. Theorem 9.13A in Rosen's Number Theory in Function Fields).

I have already proved similar statements when $k_0$ is a number field or is $\mathbb{F}_p(t)$ with $p$ an odd prime: for the latter I used the Quadratic Reciprocity Law in such fields. In the present case I was trying to use Hasse's characteristic 2 Quadratic Reciprocity Law -- for which my only exposure is this nice note of K. Conrad -- but after some hours of fiddling around, I am having trouble making this work: in the above formulation at least, what is given is not exactly a reciprocity law -- i.e., it doesn't directly compare $[w,\ell_v)$ to $[\ell_v,w)$ but only establishes a certain periodicity relation which is, in the classical cases, equivalent to QR.

Many thanks if you can help me out with this! I really need it in order to complete revisions on a long overdue paper, so a solution will be worth an acknowledgment in the paper at the least.

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# Help wanted with Cebotarev condition in characteristic 2

Having promised a longtime collaborator that I would clear my plate to finish up some joint work of ours, I am swallowing my pride and tossing up the following technical point of function field arithmetic to the experts:

Let $k_0 = \mathbb{F}_2(t)$, and let $k(P)$ be a finite separable extension of $k_0$. Let $S$ be a finite set of places $v$ of $k_0$ (thus corresponding to monic irreducible polynomials in $\mathbb{F}_2[t]$ together with possibly the place at infinity). Consider the set of monic irreducible polynomials $w \in \mathbb{F}_2[t]$ with the following properties:

(i) Every place $v$ in $S$ splits completely in the quadratic Artin-Schreier extension of $k_0$ defined by the polynomial $X^2 + X = \frac{1}{w}$, and
(ii) $w$ splits completely in $k(P)$.

Problem: Show that there are infinitely many such $w$.

The natural strategy is to show that the set of such $w$ satisfies some Cebotarev condition -- presumably it even contains the set of all primes splitting in a certain finite Galois extension of $k_0$ -- and then apply the Cebotarev Density Theorem in this context (e.g. Theorem 9.13A in Rosen's Number Theory in Function Fields).

I have already proved similar statements when $k_0$ is a number field or is $\mathbb{F}_p(t)$ with $p$ an odd prime: for the latter I used the Quadratic Reciprocity Law in such fields. In the present case I was trying to use Hasse's characteristic 2 Quadratic Reciprocity Law -- for which my only exposure is this nice note of K. Conrad -- but after some hours of fiddling around, I am having trouble making this work: in the above formulation at least, what is given is not exactly a reciprocity law -- i.e., it doesn't directly compare $[w,\ell_v)$ to $[\ell_v,w)$ but only establishes a certain periodicity relation which is, in the classical cases, equivalent to QR.

Many thanks if you can help me out with this! I really need it in order to complete revisions on a long overdue paper, so a solution will be worth an acknowledgment in the paper at the least.