Field of constants of a splitting field

Question

Let $q$ be a prime power, $\mathbb F_q$ be the finite field of order $q$, and $f\in \mathbb F_q[x]$. How would one check if the field of constants of the splitting field of $f-t$ over $\mathbb F_q(t)$ is indeed $\mathbb F_q$?

If one computes directly the splitting field, then it is relatively easy to see of course. What I am asking is if there is direct criterion to check this condition. A sufficient criterion is finding a totally split place $t_0$, i.e. $t_0$ such that $f-t_0$ splits into linear factors.

EDIT: if $q$ is large enough, the criterion is also necessary by Chebotarev Density theorem.

The largest algebraic extension of F_q contained in the splitting field (notice t is trascendental). Take as example x^3-t for q congruent to 2 modulo 3. The field of constants of the splitting field is F_(q^2) — Reyx_0, May 29 '18 at 12:28
Reyx_o, could you write up an answer giving more details on how Chebotarev helps here? — Jyrki Lahtonen, Jun 20 '18 at 21:00

Reyx_0 · Accepted Answer · 2018-06-22T17:54:12.900

2

For the other implication mentioned in the edit: let $G$ be the Galois Group of $f-t$. The identity of $G$ is a Frobenius for some place if and only if the field of constants of the splitting field of $f-t$ is trivial, i.e. is equal to $\mathbb F_q$. Now use Chebotarev on the identity to deduce that there is a totally split place of degree 1 when $q$ is large enough.

edited Jun 22 '18 at 17:54

answered Jun 21 '18 at 21:41

Reyx_0

1,158

Ok. So $q$ needs to be large in comparison to the Galois group, right? – Jyrki Lahtonen Jun 24 '18 at 07:20
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Actually q has to be large compared with the genus of the splitting field of f-t, as the error term appearing in effective Chebotarev is the same as Hasse-Weil, which is 2g sqrt(q). (This forces in turn q to be large also with #G) – Reyx_0 Jun 24 '18 at 13:01
Oh yeah :-) I will let the bounty sit for couple days more in case somebody wants to say something mote. – Jyrki Lahtonen Jun 24 '18 at 13:21
Sure, no problem :) – Reyx_0 Jun 24 '18 at 13:44
@Reyx_0 do you have a reference for the Chebotarev density theorem over $F_q(t)$ ? – mercio Jun 25 '18 at 10:38
Rosen's book, number theory in function fields. It should be Theorem 9.13B – Reyx_0 Jun 25 '18 at 19:18

Field of constants of a splitting field

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