An Elementary Test for the Galois Group of a Quartic Polynomial

Field Theory for Function Fields of Plane Quartic Curves

“…Then we have the following facts from the field theory (cf. [2]). (i) G P ∼ = S 4 ⇔g x is irreducible over k t and ψ t / ∈ k t .…”

“…, where x 0 x 1 x 2 are the homogeneous coordinates of 2 . Putting x = x 1 /x 0 , y = x 2 /x 0 , we have thatf x t = 1 − t 4 x 4 + 1 and G P ∼ = C 4 .…”

Monogenic A_4 quartic fields

“…We recall from [6] that the resolvent cubic of a quartic polynomial The next Lemma gives a general result useful for field discriminant calculations.…”

Irreducible Quartic Polynomials with Factorizations modulo p

“…Although some authors use "quartic" and "biquadratic" as synonyms, we follow Kappe and Warren [5] (among others) and reserve the latter term for polynomials of this special type. We ask which among them are irreducible over Z yet reducible modulo p for each prime p. We also investigate in section 3 the additional property of being reducible modulo n for every integer n larger than 1.…”

“…For the sake of completeness, we appeal to [5] to show that the Galois group of f e (x) is A 4 for every integer e. As the discriminant of f e (x) is a perfect square, we know that its Galois group must be a subgroup of the alternating group The case p = 2 can be treated in a similar manner. We leave the details to the reader.…”