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.