Prove that for every prime p there exists a monic irreducibl

Prove that for every prime p, there exists a monic irreducible quadratic polynomial in
Fp [x].
Does there exist a polynomial f in Z [x] such that it\'s reduction modulo p is a irreducible quadratic polynomial for all primes p simultaneously?

Solution

. Let f(x) Z[x] be a monic irreducible polynomial of degree d. Let p Z be a prime and suppose that fp(x), the image of f(x) in Zp[x], has d distinct roots in the algebraic closure of Zp. Let K denote the splitting field of f(x) over Q and E denote the splitting field of fp(x) over Zp. Then Gal(E/Zp) is isomorphic to a subgroup of Gal(K/Q). Moreover, if fp(x) factors as a product of r irreducible polynomials over Zp of degrees n1, . . . , nr, respectively, then, regarding Gal(K/Q) as a subgroup of Sd, Gal(K/Q) contains an element that can be written as a product of r disjoint cycles of disjoint cycles of lenghts n1, . . . , nr.