Irreducible Quartic Polynomials with Factorizations modulo p

Rational points on certain families of symmetric equations

“…By reducing modppq, we see that the polynomial f pxq " x 4 ´4x 2 `2 has a root in F p . This implies that p " ˘1 mod 16; see [7]. So, if p ı ˘1 mod 16, then the ρ-Selmer group has 2-rank less than or equal to one.…”

“…Therefore, after combining ( 5) and ( 6), we deduce that n pξq 1 " ˘npξq 2 for all but finitely many ξ P H 1 pK, Gq. In particular, either (7) φ…”

Quadratic $d$-numbers

“…Let k ∈ Z ≥2 be square-free and denote g(x, N, k) := x 4 − kt N x 2 + ǫ N k 2 . The conditions upon which g splits were given by Driver, Leonard, and Williams in [15]. We include the exact statements here so the reader may easily verify the arguments to follow.…”

“…Therefore Corollary 3.1.3 is the applicable result when ǫ N = 1. There are two possible values of k based on the fact that k 2 = (−k) 2 and the note following Corollary 3.1.3 in [15] implies their uniqueness for fixed g. Their defining condition is given in the statement of the lemma. Lastly, assume ǫ N = −1 in which case −k 2 is never a perfect square (in R), so Corollary 3.1.2 is the applicable result.…”

Untitled

“…Theorem 9 implies that if we know the factorization type of a polynomial f modulo a prime p that divides neither D( f ) nor the constant coefficient of f , then we can use the order of the automorphism σ p of ‫ޚ‬ p [x]/ f to obtain information about the period length of a corresponding recursive sequence modulo p. We show in this section that we can reverse this implication for polynomials f of degree k ≤ 5, using Theorem 9 together with the following application of the discriminant due to Stickelberger, adapted from [Driver et al 2005] and [Swan 1962]. …”