A generic symmetric quintic from two factorizations #
An irreducible reduction of a monic quintic makes its Galois action transitive. A second good
reduction with factor degrees (1,1,1,2) exhibits a transposition. In prime degree these two
facts force the full symmetric group, so the polynomial has label 5T5. The result applies to
any monic integral polynomial with these two pieces of factorization evidence.
Main result #
TauCeti.hasGaloisLabel_five_four_of_factorDegrees_five_and_one_one_one_two: two factorizations certify the symmetric quintic label.
theorem
TauCeti.hasGaloisLabel_five_four_of_factorDegrees_five_and_one_one_one_two
{f : Polynomial ℤ}
{p q : ℕ}
(hf : f.Monic)
(hp : HasFactorDegrees f p {5})
(hq : HasFactorDegrees f q {1, 1, 1, 2})
:
A monic integral quintic that is irreducible at one good prime and has factor degrees
(1,1,1,2) at another good prime has Galois label 5T5.