# Frobenius automorphism

An element of a Galois group of a special type. It plays a fundamental role in class field theory. Suppose that $L$ is an algebraic extension of a finite field $K$. Then the Frobenius automorphism is the automorphism $\phi _ {L/K}$ defined by the formula $\phi _ {L/K} ( a) = a ^ {q}$ for all $a \in L$, where $q = | K |$( the cardinality of $K$). If $L/K$ is a finite extension, then $\phi _ {L/K}$ generates the Galois group $\mathop{\rm Gal} ( L/K)$. For an infinite extension $L/K$, $\phi _ {L/K}$ is a topological generator of $\mathop{\rm Gal} ( L/K)$. If $L \supset E \supset K$ and $[ E: K] < \infty$, then $\phi _ {L/E} = \phi _ {L/K} ^ {[ E: K] }$.
Suppose that $k$ is a local field with a finite residue field $\overline{k}\;$, and that $K$ is an unramified extension of $k$. Then the Frobenius automorphism $\phi _ {\overline{K}\; / \overline{k}\; }$ of the extension of residue fields can be uniquely lifted to an automorphism $\phi _ {K/k} \in \mathop{\rm Gal} ( K/k)$, called the Frobenius automorphism of the unramified extension $K/k$. Let $| \overline{k}\; | = q$, let ${\mathcal O} _ {K}$ be the ring of integers of $K$, and let $\mathfrak p$ be a maximal ideal in ${\mathcal O} _ {K}$. Then the Frobenius automorphism $\phi _ {K/k}$ is uniquely determined by the condition $\phi _ {K/k} ( a) \equiv a ^ {q}$ $\mathop{\rm mod} \mathfrak p$ for every $a \in {\mathcal O} _ {k}$. If $K/k$ is an arbitrary Galois extension of local fields, then sometimes any automorphism $\phi \in \mathop{\rm Gal} ( K/k)$ that induces a Frobenius automorphism in the sense indicated above on the maximal unramified subextension of $K$ is called a Frobenius automorphism of $K/k$.
Let $K/k$ be a Galois extension of global fields, let $\mathfrak p$ be a prime ideal of $k$, and let $\mathfrak P$ be some prime ideal of $K$ over $\mathfrak p$. Suppose also that $\mathfrak P$ is unramified in $K/k$ and that $\phi _ {\mathfrak P} \in \mathop{\rm Gal} ( K _ {\mathfrak P} /k _ {\mathfrak p} )$ is the Frobenius automorphism of the unramified extension of local fields $K _ {\mathfrak P} /k _ {\mathfrak p}$. If one identifies the Galois group $\mathop{\rm Gal} ( K _ {\mathfrak P} /k _ {\mathfrak p} )$ with the decomposition subgroup of $\mathfrak P$ in $\mathop{\rm Gal} ( K/k)$, one can regard $\phi _ {\mathfrak P}$ as an element of $\mathop{\rm Gal} ( K/k)$. This element is called the Frobenius automorphism corresponding to the prime ideal $\mathfrak P$. If $K/k$ is a finite extension, then, according to the Chebotarev density theorem, for any automorphism $\sigma \in \mathop{\rm Gal} ( K/k)$ there is an infinite number of prime ideals $\mathfrak P$, unramified in $K/k$, such that $\sigma = \phi _ {\mathfrak P}$. For an Abelian extension $K/k$, the element $\phi _ {\mathfrak P}$ depends only on $\mathfrak p$. In this case $\phi _ {\mathfrak P}$ is denoted by $( \mathfrak p , K/k)$ and is called the Artin symbol of the prime ideal $\mathfrak p$.