Identifying Frobenius elements in Galois groups (Q380365)

From the text: ``We present a method to determine Frobenius elements in arbitrary Galois extensions of global fields, which may be seen as a generalisation of Euler's criterion. It is a part of the general question how to compare splitting fields and identify conjugacy classes in Galois groups, which we will discuss as well. NEWLINENEWLINEOur main result is the following:NEWLINENEWLINETheorem 1.1. Let \(K\) be a global field and \(f(x)\in K[x]\) a separable polynomial withNEWLINEGalois group \(G\). There is a polynomial \(h(x)\in K[x]\) and polynomials \(\Gamma_C\in K[X]\) indexed by the conjugacy classes \(C\) of \(G\) such thatNEWLINENEWLINE\[ \operatorname{Frob}_{\mathfrak p}\in C \Leftrightarrow \Gamma_C \left(\operatorname{Tr}_{\frac{\mathbb F_q[x]}{f(x)} \bigl/ \mathbb F_q} (h(x)x^q)\right)CARRIAGE_RETURNNEWLINE\equiv 0 \pmod{\mathfrak p} \] NEWLINENEWLINEfor almost all primes \(\mathfrak p\) of \(K\); here \(\mathbb F_q\) is the residue field at \(\mathfrak p\).NEWLINENEWLINEThis is proved in Section 5; see Theorem 5.3. Usually one can take \(h(x) = x^2\) (see below); in particular \(\operatorname{Tr}(x^{q+2}\) then determines the conjugacy class of \(\operatorname{Frob}_{\mathfrak p}\).NEWLINENEWLINEIn Section 6 we explain how the theorem recovers classical formulas for Frobenius elements in cyclotomic and Kummer extensions. In Section 7 we give explicit examples for nonabelian Galois groups, including general cubics, general quartics and quintics with Galois group \(D_{10}\).NEWLINENEWLINEThe polynomials \(\Gamma_C\) are explicitly given byNEWLINENEWLINE\[ \Gamma_C(X) = \prod_{\sigma\in C} \left(X - \sum_{j=1}^n h(a_j)\sigma(a_j)\right). \]NEWLINENEWLINEwhere \(a_1, \ldots, a_n\) are the roots of \(f\) in some splitting field. The ``almost all primes'' in the theorem are those not dividing the denominators of the coefficients of \(f\), its leading coefficient and the resultants \(\operatorname{Res}(\Gamma_C; \Gamma_{C'})\) for \(C \ne C'\); the latter simply says that the \(\Gamma_C \bmod{\mathfrak p}\) are pairwise coprime. (This condition always fails forNEWLINEramified primes.) NEWLINENEWLINEFinally, the only constraint on the polynomial \(h\) is that the resulting \(\Gamma_C\) are coprime over \(K\). This holds for almost all \(h\), in the sense that the admissible ones of degree at most \(n -1\) form a Zariski dense open subset of \(K^n\). Also, a fixed \(h\) with \(1< \deg h <n\) (for instance \(h(x) = x^2)\) will work for almost all \(f\) that define the same field; see Section 8''.