Ferdinand Georg Frobenius - Contributions To Number Theory

Contributions To Number Theory

Frobenius introduced a canonical way of turning primes into conjugacy classes in Galois groups over Q. Specifically, if K/Q is a finite Galois extension then to each (positive) prime p which does not ramify in K and to each prime ideal P lying over p in K there is a unique element g of Gal(K/Q) satisfying the condition g(x) = xp (mod P) for all integers x of K. Varying P over p changes g into a conjugate (and every conjugate of g occurs in this way), so the conjugacy class of g in the Galois group is canonically associated to p. This is called the Frobenius conjugacy class of p and any element of the conjugacy class is called a Frobenius element of p. If we take for K the mth cyclotomic field, whose Galois group over Q is the units modulo m (and thus is abelian, so conjugacy classes become elements), then for p not dividing m the Frobenius class in the Galois group is p mod m. From this point of view, the distribution of Frobenius conjugacy classes in Galois groups over Q (or, more generally, Galois groups over any number field) generalizes Dirichlet's classical result about primes in arithmetic progressions. The study of Galois groups of infinite-degree extensions of Q depends crucially on this construction of Frobenius elements, which provides in a sense a dense subset of elements which are accessible to detailed study.