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Consider an irreducible polynomial with integer coefficients. Now
reduce it modulo a prime p. Is the polynomial still irreducible? Is it the
product of distinct linear factors? Something in between?
Say that a polynomial 'splits mod p' if modulo p, if it is the product
of distinct linear factors when viewed in F_p[x]. How can one
characterize the set of primes for which our polynomial splits
completely? Is it finite? Infinite? Does it have a density?
The answers to these questions span most of what is known as Class
Field
Theory over the rationals and will be the subject of this talk.