The Theory of Algebraic Number Fields

---

In he again captured the imagination of an international audience with his famous "23 unsolved problems" of mathematics, many of which became major areas of intensive research in this century. Some of the problems remain unresolved to this day. At the end of his career, Hilbert became engrossed in the problem of providing a logically satisfactory foundation for all of mathematics.

As a result, he developed a comprehensive program to establish the consistency of axiomatized systems in terms of a metamathematical proof theory. In , Hilbert became ill with pernicious anemiathen an incurable disease. However, because Minot had just discovered a treatment, Hilbert lived for another 18 years. Springer Shop Bolero Ozon.

The Theory of Algebraic Number Fields. Under this correspondence, equivalence classes of ultrametric places of F correspond to prime ideals of O F. However, for more general number fields, the situation becomes more involved, as will be explained below. Yet another, equivalent way of describing ultrametric places is by means of localizations of O F. By the ultrametric property T is a ring. Moreover, it contains O F.

Actually, T is just the localization of O F at the prime ideal P. Conversely, P is the maximal ideal of T. Altogether, there is a three-way equivalence between ultrametric absolute values, prime ideals, and localizations on a number field.

Ramification , generally speaking, describes a geometric phenomenon that can occur with finite-to-one maps that is, maps f: For example, the map. One says that the map is "ramified" in zero. This is an example of a branched covering of Riemann surfaces. This intuition also serves to define ramification in algebraic number theory. This ideal may or may not be a prime ideal, but, according to the Lasker—Noether theorem see above , always is given by.

Whenever one ramification index is bigger than one, the prime p is said to ramify in F. In fact, unramified morphisms of schemes in algebraic geometry are a direct generalization of unramified extensions of number fields. Ramification is a purely local property, i. The inertia group measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue fields.

The following example illustrates the notions introduced above. In order to compute the ramification index of Q x , where. Since possible values for the absolute value of the place defined by the factor h are not confined to integer powers of 23, but instead are integer powers of the square root of 23, the ramification index of the field extension at 23 is two.

The valuations of any element of F can be computed in this way using resultants. If instead we eliminate with respect to the factors g and h of f , we obtain the corresponding factors for the polynomial for y , and then the adic valuation applied to the constant norm term allows us to compute the valuations of y for g and h which are both 1 in this instance. Much of the significance of the discriminant lies in the fact that ramified ultrametric places are all places obtained from factorizations in Q p where p divides the discriminant.

This is even true of the polynomial discriminant; however the converse is also true, that if a prime p divides the discriminant, then there is a p -place which ramifies. For this converse the field discriminant is needed. This is the Dedekind discriminant theorem. The Dedekind discriminant tells us it is the only ultrametric place which does.

The other ramified place comes from the absolute value on the complex embedding of F. This is the first stepstone into Iwasawa theory. The fundamental theorem of Galois theory links fields in between F and its algebraic closure and closed subgroups of Gal F. For example, the abelianization the biggest abelian quotient G ab of G corresponds to a field referred to as the maximal abelian extension F ab called so since any further extension is not abelian, i.

By the Kronecker—Weber theorem , the maximal abelian extension of Q is the extension generated by all roots of unity. For more general number fields, class field theory , specifically the Artin reciprocity law gives an answer by describing G ab in terms of the idele class group.

Also notable is the Hilbert class field , the maximal abelian unramified field extension of F. It can be shown to be finite over F , its Galois group over F is isomorphic to the class group of F , in particular its degree equals the class number h of F see above.

Algebraic number field

In certain situations, the Galois group acts on other mathematical objects, for example a group. Such a group is then also referred to as a Galois module. This enables the use of group cohomology for the Galois group Gal F , also known as Galois cohomology , which in the first place measures the failure of exactness of taking Gal F -invariants, but offers deeper insights and questions as well. This Galois module plays a significant role in many arithmetic dualities , such as Poitou-Tate duality.

Generally speaking, the term "local to global" refers to the idea that a global problem is first done at a local level, which tends to simplify the questions. Then, of course, the information gained in the local analysis has to be put together to get back to some global statement. For example, the notion of sheaves reifies that idea in topology and geometry.

Number fields share a great deal of similarity with another class of fields much used in algebraic geometry known as function fields of algebraic curves over finite fields. An example is F p T. They are similar in many respects, for example in that number rings are one-dimensional regular rings, as are the coordinate rings the quotient fields of which is the function field in question of curves. Therefore, both types of field are called global fields. In accordance with the philosophy laid out above, they can be studied at a local level first, that is to say, by looking at the corresponding local fields.

For number fields F , the local fields are the completions of F at all places, including the archimedean ones see local analysis. For function fields, the local fields are completions of the local rings at all points of the curve for function fields. Many results valid for function fields also hold, at least if reformulated properly, for number fields.

However, the study of number fields often poses difficulties and phenomena not encountered in function fields. For example, in function fields, there is no dichotomy into non-archimedean and archimedean places. Nonetheless, function fields often serves as a source of intuition what should be expected in the number field case. A prototypical question, posed at a global level, is whether some polynomial equation has a solution in F. If this is the case, this solution is also a solution in all completions. The local-global principle or Hasse principle asserts that for quadratic equations, the converse holds, as well.

Thereby, checking whether such an equation has a solution can be done on all the completions of F , which is often easier, since analytic methods classical analytic tools such as intermediate value theorem at the archimedean places and p-adic analysis at the nonarchimedean places can be used. This implication does not hold, however, for more general types of equations. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers , rational numbers , and their generalizations.

Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers , finite fields , and function fields. 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.

The beginnings of algebraic number theory can be traced to Diophantine equations, [1] named after the 3rd-century Alexandrian mathematician, Diophantus , who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B , respectively:. Diophantine equations have been studied for thousands of years.

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. 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.

Navigation menu

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. 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.

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.

Algebraic number theory

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

• Class 2 Transferases II: 29 (Springer Handbook of Enzymes)
• The Adventure of the Apprentices Coin
• The Rest of Us: Dispatches from the Mother Ship
• Dragon Rescue
• Romantisches Spiel in Griechenland (ROMANA) (German Edition)
• Something in the Sea
• Beyond the Phantom Battle: Mystery at Loch Ashie