Frobenius in a small cyclotomic field

Given: The data and hypotheses of the statement.

Frobenius cycle type and prime splitting: Let $F\in \mathbb{Z}[T]$ be monic separable with splitting field L, and let p be a rational prime not dividing $\mathrm{Disc}(F)$. Then p is unramified in L and $\bar{F}\in {\mathbb{F}}_p[T]$ is squarefree. The degrees of its monic irreducible factors, with each distinct factor counted once, are exactly the cycle lengths of arithmetic Frobenius on the roots of F.

Frobenius order is residue degree: For finite Galois L/K and nonzero $P\mid p$, the arithmetic Frobenius coset has order $f(P\mathrm{/}p)$ in D/I. If P is unramified, ${\mathrm{Frob}}_P$ has the same order in D.

Galois prime decomposition efg: For a finite Galois extension L/K and nonzero prime p, every P above p has the same ramification index e and residue degree f. If there are g such primes, then $efg=[L:K]$.

Eisenstein criterion over the integers: Let $f=a_n{x}^n+\cdots +a_0\in \mathbb{Z}[x]$ be primitive with $n\ge 1$. If there is a prime $p$ such that $p\nmid a_n,p\mid a_i\text{ for every }i<n,p^2\nmid a_0,$ then $f$ is irreducible in $\mathbb{Q}[x]$.

The discriminant of a monic polynomial as the coefficient expression of ${\mathrm{\Delta }}_n^2$: By prop-vandermonde-square-is-symmetric and thm-fundamental-theorem-of-symmetric-polynomials, there is a unique polynomial $D_n\in \mathbb{Z}[T_1,\dots ,T_n]$ such that ${\mathrm{\Delta }}_n(x_1,\dots ,x_n{)}^2=D_n(e_1,\dots ,e_n)\mathrm{.}$ For a monic polynomial $f(t)=t^n+a_1{t}^{n-1}+\cdots +a_n$ over a commutative ring, its discriminant is $\mathrm{Disc}(f):=D_n(-a_1,a_2,\dots ,(-1{)}^n{a}_n)\mathrm{.}$ Equivalently, in any algebra in which $f$ splits with roots ${\alpha }_1,\dots ,{\alpha }_n$, this coefficient expression evaluates to ${\mathrm{\Delta }}_n({\alpha }_1,\dots ,{\alpha }_n{)}^2$. The definition therefore depends only on the coefficients and not on a choice or ordering of roots. For a monic constant polynomial, $\mathrm{Disc}(1)=1$.