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Heritage of European Mathematics Peter Roquette (University of Heidelberg, Germany). Contributions to the History of Number Theory in the 20th Century.
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Diophantus' major work was the Arithmetica , of which only a portion has survived. Fermat's last theorem was first conjectured by Pierre de Fermat in , famously in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin.

Paul Erdős

No successful proof was published until despite the efforts of countless mathematicians during the intervening years. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. One of the founding works of algebraic number theory, the Disquisitiones Arithmeticae Latin : Arithmetical Investigations is a textbook of number theory written in Latin [4] by Carl Friedrich Gauss in when Gauss was 21 and first published in when he was In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler , Lagrange and Legendre and adds important new results of his own.

Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways. Many of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished.

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 and Peter Gustav Lejeune Dirichlet proved the first class number formula , for quadratic forms later refined by his student Leopold Kronecker.

The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general number fields. He first used the pigeonhole principle , a basic counting argument, in the proof of a theorem in diophantine approximation , later named after him Dirichlet's approximation theorem.

Richard Dedekind 's study of Lejeune Dirichlet's work was what led him to his later study of algebraic number fields and ideals. The word "Ring", introduced later by Hilbert , does not appear in Dedekind's work. Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients.

The concept underwent further development in the hands of Hilbert and, especially, of Emmy Noether. David Hilbert unified the field of algebraic number theory with his treatise Zahlbericht literally "report on numbers". He also resolved a significant number-theory problem formulated by Waring in 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. He made a series of conjectures on class field theory.

The concepts were highly influential, and his own contribution lives on in the names of the Hilbert class field and of the Hilbert symbol of local class field theory. Results were mostly proved by , after work by Teiji Takagi. Emil Artin established the Artin reciprocity law in a series of papers ; ; This law is a general theorem in number theory that forms a central part of global class field theory. Artin's result provided a partial solution to Hilbert's ninth problem. Around , Japanese mathematicians Goro Shimura and Yutaka Taniyama observed a possible link between two apparently completely distinct, branches of mathematics, elliptic curves and modular forms.

The resulting modularity theorem at the time known as the Taniyama—Shimura conjecture states that every elliptic curve is modular , meaning that it can be associated with a unique modular form. It became a part of the Langlands program , a list of important conjectures needing proof or disproof. From to , Andrew Wiles provided a proof of the modularity theorem for semistable elliptic curves , which, together with Ribet's theorem , provided a proof for Fermat's Last Theorem.

Almost every mathematician at the time had previously considered both Fermat's Last Theorem and the Modularity Theorem either impossible or virtually impossible to prove, even given the most cutting edge developments.

Biographies of past number theorists and various items of historical interest

Wiles first announced his proof in June [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 , and formally published in 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.

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An important property of the ring of integers is that it satisfies the fundamental theorem of arithmetic , that every positive integer has a factorization into a product of prime numbers , and this factorization is unique up to the ordering of the factors. This may no longer be true in the ring of integers O of an algebraic number field K. A prime element is an element p of O such that if p divides a product ab , then it divides one of the factors a or b. This property is closely related to primality in the integers, because any positive integer satisfying this property is either 1 or a prime number.

However, it is strictly weaker.

If factorizations into prime elements are permitted, then, even in the integers, there are alternative factorizations such as. In general, if u is a unit , meaning a number with a multiplicative inverse in O , and if p is a prime element, then up is also a prime element. Numbers such as p and up are said to be associate. Requiring that prime numbers be positive selects a unique element from among a set of associated prime elements.

When K is not the rational numbers, however, there is no analog of positivity. This leads to equations such as.

The History of Mathematics and Its Applications

For this reason, one adopts the definition of unique factorization used in unique factorization domains UFDs. 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.

There is an algebraic obstruction called the ideal class group. When the ideal class group is trivial, the ring is a UFD.

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When it is not, there is a distinction between a prime element and an irreducible element. These are the elements that cannot be factored any further. Every element in O admits a factorization into irreducible elements, but it may admit more than one.


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This is because, while all prime elements are irreducible, some irreducible elements may not be prime. This means that the number 9 has two factorizations into irreducible elements,. Unlike the situation with units, where uniqueness could be repaired by weakening the definition, overcoming this failure requires a new perspective. If I is an ideal in O , then there is always a factorization. In particular, this is true if I is the principal ideal generated by a single element. This is the strongest sense in which the ring of integers of a general number field admits unique factorization.

In the language of ring theory, it says that rings of integers are Dedekind domains. When O is a UFD, every prime ideal is generated by a prime element.


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  5. Otherwise, there are prime ideals which are not generated by prime elements. Historically, the idea of factoring ideals into prime ideals was preceded by Ernst Kummer's introduction of ideal numbers. These are numbers lying in an extension field E of K. This extension field is now known as the Hilbert class field. By the principal ideal theorem , every prime ideal of O generates a principal ideal of the ring of integers of E. A generator of this principal ideal is called an ideal number.

    Kummer used these as a substitute for the failure of unique factorization in cyclotomic fields. These eventually led Richard Dedekind to introduce a forerunner of ideals and to prove unique factorization of ideals.

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    17th Century Mathematics - The Story of Mathematics

    An ideal which is prime in the ring of integers in one number field may fail to be prime when extended to a larger number field. Consider, for example, the prime numbers. The corresponding ideals p Z are prime ideals of the ring Z. However, when this ideal is extended to the Gaussian integers to get p Z [ i ] , it may or may not be prime. 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.

    Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory. Class field theory accomplishes this goal when K is an abelian extension of Q i. Unique factorization fails if and only if there are prime ideals that fail to be principal.