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 $.

References

[1]	A. Weil, "Basic number theory" , Springer (1974)