EUDML

Let $Q$ denote the field of rational numbers. Let $K$ be a cyclic quartic extension of $Q$ . It is known that there are unique integers $A$ , $B$ , $C$ , $D$ such that $K = Q (\sqrt{A (D + B \sqrt{D})}),$ where $A is squarefree and odd, D = B^{2} + C^{2} is squarefree, B > 0, C > 0, G C D (A, D) = 1 .$ The conductor $f (K)$ of $K$ is $f (K) = 2^{l} | A | D$ , where $l = \{\begin{matrix} 3, if D \equiv 2 (mod 4) or D \equiv 1 (mod 4), B \equiv 1 (mod 2), \\ 2, if D \equiv 1 (mod 4), B \equiv 0 (mod 2), A + B \equiv 3 (mod 4), \\ 0, if D \equiv 1 (mod 4), B \equiv 0 (mod 2), A + B \equiv 1 (mod 4) . \end{matrix}$ A simple proof of this formula for $f (K)$ is given, which uses the basic properties of quartic Gauss sums.

Predictive criteria for the representation of primes by binary quadratic forms

Joseph B. Muskat; Blair K. Spearman; Kenneth S. Williams — 1995

Acta Arithmetica

Pascal's triangle (mod 9)

James G. Huard; Blair K. Spearman; Kenneth S. Williams — 1997

Acta Arithmetica

On the prime factors of non-congruent numbers

Lindsey Reinholz; Blair K. Spearman; Qiduan Yang — 2015

Colloquium Mathematicae

We give infinitely many new families of non-congruent numbers where the first prime factor of each number is of the form 8k+1 and the rest of the prime factors have the form 8k+3. Products of elements in each family are shown to be non-congruent.

PSL $(2, 7)$ septimic fields with a power basis

Melisa J. Lavallee; Blair K. Spearman; Qiduan Yang — 2012

Journal de Théorie des Nombres de Bordeaux

We give an infinite set of distinct monogenic septimic fields whose normal closure has Galois group $P S L (2, 7)$ .

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On the Lagrange resolvents of a dihedral quintic polynomial.

A relative integral basis over $ℚ (\sqrt{- 3})$ for the normal closure of a pure cubic field.

On a sequence of nonsolvable quintic polynomials.

Explicit decomposition of a rational prime in a cubic field.

On the common index divisors of a dihedral field of prime degree.

Cyclotomy of order 15 over $G F (p^{2})$ , $p = 4, 11$ (mod 15).

Normal integral bases for Emma Lehmer’s parametric family of cyclic quintics

The conductor of a cyclic quartic field using Gauss sums

Predictive criteria for the representation of primes by binary quadratic forms

Pascal's triangle (mod 9)

On the prime factors of non-congruent numbers

PSL $(2, 7)$ septimic fields with a power basis