Algebraic number theory

These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

For example, the solutions to the quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC).

[2] Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC).

Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems and conjectures.

The Disquisitiones was the starting point for the work of other nineteenth century European mathematicians including Ernst Kummer, Peter Gustav Lejeune Dirichlet and Richard Dedekind.

They must have appeared particularly cryptic to his contemporaries; we can now read them as containing the germs of the theories of L-functions and complex multiplication, in particular.

In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, for quadratic forms (later refined by his student Leopold Kronecker).

He published important contributions to Fermat's last theorem, for which he proved the cases n = 5 and n = 14, and to the biquadratic reciprocity law.

(Edwards 1983)1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory.

As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers.

[7] He then had little more to publish on the subject; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.

Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama observed a possible link between two apparently completely distinct, branches of mathematics, elliptic curves and modular forms.

It became a part of the Langlands program, a list of important conjectures needing proof or disproof.

Wiles first announced his proof in June 1993[11] in a version that was soon recognized as having a serious gap at a key point.

The proof was corrected by Wiles, partly in collaboration with Richard Taylor, and the final, widely accepted version was released in September 1994, and formally published in 1995.

The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics.

It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat.

In a UFD, the prime elements occurring in a factorization are only expected to be unique up to units and their ordering.

However, even with this weaker definition, many rings of integers in algebraic number fields do not admit unique factorization.

This means that the number 9 has two factorizations into irreducible elements, This equation shows that 3 divides the product (2 + √-5)(2 - √-5) = 9.

Unlike the situation with units, where uniqueness could be repaired by weakening the definition, overcoming this failure requires a new perspective.

This is the strongest sense in which the ring of integers of a general number field admits unique factorization.

A complete answer to the question of which ideals remain prime in the Gaussian integers is provided by Fermat's theorem on sums of two squares.

Real and complex embeddings can be put on the same footing as prime ideals by adopting a perspective based on valuations.

Therefore, absolute values are a common language to describe both the real embedding of Q and the prime numbers.

Reinterpreting this in the context of a number field, the torsion part consists of the roots of unity that lie in O.

Therefore, one localizes the ring of algebraic integers to that prime and then completes the fraction field much in the spirit of geometry.

This is a consequence of Minkowski's theorem since there are only finitely many Integral ideals with norm less than a fixed positive integer[15] page 78.

In other words, O× is a finitely generated abelian group of rank r1 + r2 − 1 whose torsion consists of the roots of unity in O.

The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (p/q) generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between (p/q) and (q/p).