He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms.

In 1874, after having taught at secondary school level first at the Joachimsthal Gymnasium then at the Sophienrealschule, he was appointed to the University of Berlin as an extraordinary professor of mathematics.

[2] Frobenius was only in Berlin a year before he went to Zürich to take up an appointment as an ordinary professor at the Eidgenössische Polytechnikum.

It was there that he married, brought up his family, and did much important work in widely differing areas of mathematics.

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.