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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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Weil,   "Basic number theory" , Springer  (1974) {{ZBL|0326.12001}} </TD></TR></table>