Mathematics 676-4 23) Homework (due Mar. 5) A. Hulpke Factorize the polynomial x8 − x7 + 7x6 − 6x5 + 15x4 − 9x3 + 40x2 − 31x + 31 using the “big-prime” method with modular factorization and combination of factors. 24∗ ) Why does one not consider a “small-prime” approach (calculate the result modulo several primes and combine using the Chinese Remainder theorem) for the factorization of polynomials in Z[x]? 25) Factorize the polynomial f = x6 − 4x2 − 1 ∈ Q[x] modulo all primes < 1000 that do not divide disc( f ) and collect the occurring degree distributions. Based on this, what is Gal( f ) likely to be? 26) a) Let p be a prime and G ≤ S p with (1, 2, 3, . . . , p) ∈ G, (i, j) ∈ p (for some 1 ≤ i, j ≤ p). Show that G = S p . b) Construct a polynomial with Galois group S7 . (Hint: Frobenius’ theorem.) 27) a) Show that (in the limit for n → ∞) 1n of all polynomials of degree n over F p are irreducible. b) Let f ∈ Z[x] be a random polynomial. Show that f will be reducible modulo k different primes with probability (1 − n1 )k . c) Show that a random polynomial f ∈ Z[x] is with probability 1 irreducible. With a similar argument one can show (B.L. V. D .WAE RDE N, Die Seltenheit der Gleichungen mit Affekt, Math. Ann. 109 (1934), 13–16) that asymptotically 100% of all polynomials have Galois group An or Sn . Problems marked with ∗ are bonus problems for extra credit.