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1. Number of solutions of congruences. Let f(x) = x + a1x d−1 + · · · + ad, a1, . . . , ad ∈ Z be an irreducible polynomial. Let Nf (n) be the number of solutions of f(x) ≡ 0 (modn). It is an important problem to study Nf (n). Let L be the splitting field of f with the Galois group G. Let E = Q[α], where α is a root of f . Then [E : Q] = d. Let Gal(L/E) = H. Let S(L/Q) = {p : Nf (p) = d}. Then it is known that S(L/Q) determines L completely. It is a goal of the class field theory to determine the set S(L/Q). Except for finitely many primes, p ∈ S(L/Q) if and only if p splits completely in L: this comes from the fact that p splits completely in L if and only if p splits completely in E. It is clear that if p splits completely in L, then p splits completely in E. Conversely, let P, p be primes in L,E, respectively, such that P | p, p | p. Then p splits completely in L if and only if LP = Qp. We fix an embedding Qp →֒ Qp. Suppose p splits completely in E. Then for any p | p, E ⊂ Ep = Qp →֒ Qp. So every conjugate of E is contained in Qp. Hence L is contained in Qp. So p splits completely in L. By the well-known theorem of Dedekind (e.g. [N, Theorem 4.33]), except for finitely many primes (in fact, if p does not divide the discriminant of f(x), or (OE : Z[α])), p splits completely in E if and only if Nf (p) = d. Consider IndH 1 = 1 + ̺, where ̺ : G → GLd−1(C) is a direct sum of non-trivial irreducible representations of G, i.e.,

Functoriality and number of solutions of congruences