This paper gives a basic introduction to Minkowski Theory and the class group, leading up to a proof that the class number (the order of the class group) is finite. This paper is based on Jürgen Neukirch’s Algebraic Number Theory, but provides more detailed proofs and explanations as well as numerous examples. The goal of this paper is to present the concepts in Neukirch’s book in such a way th...

We specialize the Eichler-Selberg trace formula to obtain trace formulas for the prime-to-level Hecke action on cusp forms for certain congruence groups of arbitrary level. As a consequence, we determine the asymptotic in the prime p of the number of (weighted) SL2(Z)-conjugation orbits of 2 × 2 matrices of determinant p whose reductions modulo N lie in a given conjugacy class in GL2(Z/NZ). Thi...

Proof. Using that (Z/p)× is a cyclic group of order p − 1 (i.e. the existence of primitive roots), we see that there is a square root of −1 (that is, a non-trivial fourth root of 1) in (Z/p)× if and only if p ≡ 1 mod 4. Suppose now that p ≡ −1 mod 4, and suppose that α and β are two elements of Z[i] such that p|αβ. Then p = N(p)|N(α)N(β), and so (after relabelling if necessary) we may assume th...

1. Let g denote any odd prime and h = h(g) the class number of the cyclotomic field R(r), where r is the primitive gth root of unity, R the rational numbers. It is known that we can write: h = h1h2, where hi and h2 (both integers) are the so-called first and second factors of the class number; in fact h2 is the class number of the real field of degree 2 under R(r), namely the field R(D + D-). K...

Assuming the 2-adic Iwasawa main conjecture, we find all CM fields with higher relative class number at most 16: there are at least 31 and at most 34 such fields, and exactly one is not abelian. The problem of determining all imaginary quadratic fields K = Q( √ d) of small class number h(K) was first posed in Article 303 of Gauss’ Disquisitiones Arithmeticae. It would take almost 150 years of w...

We improve a criterion of Inkeri and show that if there is a solution to Catalan’s equation x − y = ±1, (1) with p and q prime numbers greater than 3 and both congruent to 3 (mod 4), then p and q form a double Wieferich pair. Further, we refine a result of Schwarz to obtain similar criteria when only one of the exponents is congruent to 3 (mod 4). Indeed, in light of the results proved here it ...

After a review of the quadratic case, a general problem about the existence of number fields of a fixed degree with extremely large class numbers is formulated. This problem is solved for abelian cubic fields. Then some conditional results proven elsewhere are discussed about totally real number fields of a fixed degree, each of whose normal closure has the symmetric group as Galois group.