Algebraic number theory

The branch of number theory with the basic aim of studying properties of algebraic integers in algebraic number fields $ K $ of finite degree over the field $ \mathbf Q $ of rational numbers (cf. Algebraic number).

The set of algebraic integers $ O _{K} $ of a field $ K / \mathbf Q $ — an extension $ K $ of $ \mathbf Q $ of degree $ n $ (cf. Extension of a field) — can be obtained from an integral basis $ ( \omega _{1} \dots \omega _{n} ) $ ; this means that each algebraic integer (i.e. element of $ O _{K} $ ) can be written in the form $ x _{1} \omega _{1} + \dots + x _{n} \omega _{n} $ where all the $ x _{i} $ run through the rational integers (i.e. $ \mathbf Z $ ). Moreover, such a representation is unique for each algebraic integer in $ K $ .

However, properties of rational integers often do not have obvious analogues for algebraic integers. The first such property is related to units, the invertible elements of $ O _{K} $ (cf. Unit). The field of rational numbers has only $ +1 $ and $ -1 $ as units, but a general algebraic number field may contain an infinite number of units. E.g. consider the real quadratic field $ \mathbf Q ( \sqrt D ) $ , where $ D > 1 $ is a rational integer not equal to a square. If, moreover, $ D \not\equiv 1 $ $ ( \mathop{\rm mod}\nolimits \ 4 ) $ , then $ ( 1 ,\ \sqrt D ) $ is an integral basis for it. The Pell equation $ x ^{2} - D y ^{2} = 1 $ has an infinite number of solutions $ ( x ,\ y ) \in \mathbf Z \times \mathbf Z $ . Any of them gives rise to a unit $ x + y \sqrt D $ of $ \mathbf Q ( \sqrt D ) $ . In fact, $$ ( x + y \sqrt D ) ( x - y \sqrt D ) = 1 , $$ and $$ \frac{1}{x + y \sqrt D} = x - y \sqrt D $$ is also an algebraic integer in $ \mathbf Q ( \sqrt D ) $ . The units of $ \mathbf Q ( \sqrt D ) $ form an infinite multiplicative group (the group of Pell units). The question arises: What is the structure of this group?

A second property of rational numbers without an obvious analogue for algebraic numbers is related to the theorem on unique factorization of rational integers $ n $ in prime factors: $$ n = p _{1} ^ {a _{1}} \dots p _{k} ^ {a _{k}} . $$ For algebraic numbers such a unique factorization need not hold. E.g. consider the field $ \mathbf Q ( \sqrt -5 ) $ . In it, the number 6 has two essentially different factorizations: $ 6 = 2 \cdot 3 $ , $ 6 = ( 1 + \sqrt -5 ) ( 1 - \sqrt -5 ) $ . For extensions of higher degree the situation becomes still more complicated. The question arises: What becomes of the unique factorization theorem, does it have a meaning at all in algebraic number fields?

A third property without an obvious analogue is related to prime numbers. An ordinary prime number need not remain a prime number in an algebraic number field. E.g. in the field $ \mathbf Q ( \sqrt -1 ) $ of Gaussian numbers the prime number 5 has a factorization: $ 5 = ( 2 + \sqrt -1 ) ( 2 - \sqrt -1 ) $ . However, the prime number 7 remains prime in this field. The question arises: Do there exist general laws governing the behaviour of prime numbers in algebraic number fields of higher degree? In other words, is it possible to find rules that would give a definite answer to the following question: Does a given prime number remain prime in a field $ K / \mathbf Q $ or does it split in it, and if it splits, in how many factors?

Finally, a last (fourth) problem concerns the general structure of algebraic number fields. The field $ \mathbf Q $ is the minimal field of characteristic zero without proper subfields. Any other algebraic number field does have subfields. E.g., $ \mathbf Q $ is a subfield of any algebraic number field. The question arises: How many subfields are contained in a given extension $ K / \mathbf Q $ , finitely or infinitely many, and what is their structure?

These are four main problems in algebraic number theory, and answering them constitutes the content of algebraic number theory. It is quite natural to start with the fourth problem, since its answer will shed light on the other three. The problem was solved by E. Galois in the 1820s (cf. Galois theory). The fact that the number of subfields of an extension $ K / \mathbf Q $ of degree $ n $ over $ \mathbf Q $ is finite follows from the existence of a one-to-one correspondence (the fundamental Galois correspondence) between all subfields of the normal closure of $ K $ (cf. Normal extension) and the subgroups of its Galois group, a finite group of finite order (number of elements) at most $ n! $ .

The structure of the group of units of a field was elucidated by P. Dirichlet. His basic idea can be given by taking the group of Pell units (cf. above) as example. Any power of such a unit (both positive or negative) is a unit. There is a fundamental unit $ \eta = x _{0} + y _{0} \sqrt D $ , and all other units are plus or minus integral powers of it; hence, the Pell units form the product of $ \langle -1 \rangle $ with an infinite cyclic group. This fact is a special case of the general Dirichlet unit theorem for algebraic number fields: If the degree of a field $ K / \mathbf Q $ is $ n = r _{1} + 2 r _{2} $ , where $ r _{1} $ is the number of real imbeddings $ K \rightarrow \mathbf R $ and $ r _{2} $ the number of complex-conjugate pairs of complex imbeddings $ K \rightarrow \mathbf C $ , then the group of units $ U $ of $ K $ is the product of a finite cyclic group $ C $ and $ r = r _{1} + r _{2} -1 $ infinite cyclic groups: $$ U = C \times \langle \eta _{1} \rangle \times \dots \times \langle \eta _{r} \rangle . $$ Here $ \eta _{1} \dots \eta _{r} $ are independent fundamental units and $ C $ is the group of roots of unity contained in $ K $ . The norm of any unit in a field, i.e. the product of the unit by its conjugates, is equal to $ \pm 1 $ .

The problem of the non-unique factorization of algebraic integers in algebraic number fields was solved by E. Kummer, who started from a special case; he tried to solve Fermat's last theorem on the impossibility of solving the equation $ x ^{p} + y ^{p} = z ^{p} $ in non-zero integers for any prime number $ p > 2 $ . Kummer expanded the left-hand side using $ p $ -th roots of unity, and hence the problem was reduced to one about the algebraic integers of $ \mathbf Q ( \zeta ) $ , $ \zeta $ a primitive $ p $ -th root of unity. If there would be unique factorization into prime factors in $ \mathbf Q ( \zeta ) $ , then it would have been sufficient to prove that not all prime factors at the left-hand side occur with an exponent that is a multiple of $ p $ . This was Kummer's first point of view, but Dirichlet pointed out to him that unique factorization need not hold. In order to overcome this difficulty Kummer introduced ideal numbers, thus altering the entire structure of algebraic number theory for the future. The concept of an ideal number arises from the fact that if a field $ k $ does not contain prime numbers into which any algebraic integer in $ k $ can be split, then there is a field $ K / k $ of finite degree over $ k $ in which there does exist a collection of numbers that play the role of primes for $ k $ . These numbers were called ideal by Kummer (since they do not lie in the original field $ k $ ). By introducing ideal numbers the theorem on unique factorization in $ k $ holds. Two numbers in a field that differ by a unit (so-called associated numbers) have one and the same ideal factors. (Note that ideal numbers are defined relative to the original field $ k $ ; for another field $ k ^ \prime $ one must construct an extension $ K ^ \prime / k ^ \prime $ (of possibly different degree over $ k ^ \prime $ ) in which all ideal numbers of $ k ^ \prime $ are contained.)

Kummer also introduced the concept of the ideal class number of a field $ k $ . Two ideal numbers are said to belong to the same class if their quotient belongs to $ k $ . The number of classes thus obtained is called the ideal class number of $ k $ . He obtained the following important result: The class number $ h $ of $ k $ is finite, and the classes form an Abelian group under multiplication. Thus, any ideal number can be regarded as the $ h $ -th root of some element of the original field $ k $ . The class number can be explicitly described in terms of other field constants (the regulator, the discriminant and the degree of the field).

Subsequently, the concept of an ideal number was replaced by that of an ideal, equivalent to it, which can be described in terms of the field $ k $ itself. Already in the 1950s the concept of an ideal was generalized to the more comprehensive concept of a divisor. Therefore, the modern theory of Kummer may be stated in terms of divisors. For algebraic number fields, however, the concepts of a divisor and an ideal coincide, and such fields only will be considered in the sequel. The concept of an ideal is related to that of non-associated numbers, which helps in understanding the deep connections between Kummer's theory and Dirichlet's theory of units. Although Kummer did not succeed in solving Fermat's problem, his ideas extended far beyond this problem and the concept of an ideal has now become fundamental in many branches of mathematics.

Since prime ideal numbers are defined relative to a field or, in modern terminology, since they are prime ideals in a field, the third problem on the factorization of ordinary prime numbers in an algebraic number field can be stated in the following general form. Suppose one is given a field $ k $ and a prime ideal $ \mathfrak p $ of its ring of algebraic integers. The question is: Does $ \mathfrak p $ remain prime in an extension $ K = k ( \theta ) $ or does it split in a product of prime ideals of the ring of algebraic integers of $ K $ ? In the latter case: By which law does it split? This question led to class field theory, a central part of modern algebraic number theory. The first solution of this problem was given by Kummer, who proved that if $ \theta $ is a root of an irreducible polynomial $ f ( x ) $ and $ 1 ,\ \theta, \dots, \theta ^{n-1} $ is an integral basis for $ O _{K} $ over $ O _{k} $ , then $ \mathfrak p $ splits in $ k ( \theta ) $ "by the same law as $f(x)$" does in the residue field ( $ \mathop{\rm mod}\nolimits \ \mathfrak p $ ). In other words, the factorization of $ \mathfrak p $ in $ k ( \theta ) $ is determined by the congruence $$ f (x) \equiv f _{1} ^ {\ \alpha _{1}} (x) \dots f _{m} ^ {\ \alpha _{m}} (x) ( \mathop{\rm mod}\nolimits \ \mathfrak p ) . $$ The corresponding factorization is called Kummer's formula (or Kummer's decomposition) $$ \tag{1} \mathfrak p = \mathfrak P _{1} ^ {\alpha _{1}} \dots \mathfrak P _{m} ^ {\alpha _{m}} , $$ where $ \mathfrak P _{i} $ are prime ideals in $ K = k ( \theta ) $ .

In principle, this equation solves the third problem in algebraic number theory, but it is a local equation in the sense that it is necessary to check it for each prime ideal separately. The problem of partitioning the set of all prime ideals in classes such that the factorization law is the same in any given class and such that, moreover, it is possible to give a simple description of these classes, is solved by class field theory for extensions $ K / k $ with Abelian Galois group $ \mathop{\rm Gal}\nolimits ( K / k ) $ .

Equation (1) yields a preliminary concept of a class. Let $ n $ be the degree of the extension $ K / k $ and let $ f _{i} $ be the relative degree of the ideal $ \mathfrak P _{i} $ . Computing the relative norms $ N _ {K / k} $ of both parts of (1) leads to $$ \tag{2} n = a _{1} f _{1} + \dots + a _{m} f _{m} , $$ in which $ a _{i} $ and $ f _{i} $ are natural numbers. For $ n $ fixed equation (2) has a finite number of solutions, so that the set of all prime ideals of $ k $ can be partitioned into a finite number of classes and one can collect the prime ideals whose Kummer decomposition corresponds to one solution $ ( a _{i} ,\ f _{i} ) $ of (2) into one class. The number of prime ideals with the property that some $ a _{i} > 2 $ , is finite, and they all are divisors of the discriminant $ \mathfrak B $ of $ K / k $ . Only infinite classes are of interest, so that classes for which $ a _{i} \geq 2 $ can be ignored.

In order to simplify the exposition, the field $ K / k $ is considered to be normal from now on. In such fields the condition $$ f _{1} = \dots = f _{m} $$ holds. Therefore, the set of all $ \mathfrak p $ not dividing $ \mathfrak B $ is partitioned into $ d (n) $ classes, where $ d (n) $ denotes the number of divisors. The class with $ m = n $ is of special interest, in it $$ f _{1} = \dots = f _{m} = 1 . $$ A prime ideal $ \mathfrak p $ in it has maximal number of prime divisors in $ K / k $ : $$ \tag{3} \mathfrak p = \mathfrak p _{1} \dots \mathfrak p _{n} . $$ Such $ \mathfrak p $ are said to split completely (or to be totally split), and their class is called the principal class of $ k $ relative to $ K / k $ . It is a principal object of study in class field theory. In order to be able to define the principal class via (3) it is necessary to give a proof of the facts that prime ideals satisfying (1) do exist in $ k $ and that there are infinitely many such ideals. Therefore a basic problem in class field theory is to define the principal class in terms of the field $ k $ itself in such a way that its infinite nature would follow. This problem has been completely solved for Abelian extensions $ K / k $ .

In order to be able to understand the ideas of class field theory better, it is necessary to introduce the general concept of an ideal class group. Kummer's definition given above then corresponds to the modern concept of an absolute ideal class group. The general concept of an ideal class group which is used nowadays is due to H. Weber and T. Takagi, [5].

Weber introduced the concept of the conductor of a class group. Let $ \mathfrak m $ be an integral ideal of $ k $ , let $ L _{\mathfrak m} $ be the subgroup of principal ideals $ ( \alpha ) $ of $ k $ given by $ \alpha \equiv 1 $ $ ( \mathop{\rm mod}\nolimits \ \mathfrak m ) $ and let $ A _{\mathfrak m} $ be the subgroup of all ideals of $ k $ that are relatively prime with $ \mathfrak m $ . A subgroup $ H _{\mathfrak m} $ satisfying $ L _{\mathfrak m} \subseteq H _{\mathfrak m} \subseteq A _{\mathfrak m} $ can be regarded as a group of principal ideals, and a (generalized) class group $ A _{\mathfrak m} / H _{\mathfrak m} $ is constructed in this way. For $ \mathfrak m = 1 $ and $ H _{1} = L _{1} $ one obtains Kummer's definition. In the general case $ H _{\mathfrak m} $ consists of progressions $ ( \mathop{\rm mod}\nolimits \ \mathfrak m ) $ whose residue classes form a subgroup of the whole multiplicative group $ ( \mathop{\rm mod}\nolimits \ \mathfrak m ) $ . The order $ h _{\mathfrak m} $ of this generalized class group satisfies the estimates $ 1 <= h _{\mathfrak m} <= h \cdot \phi ( \mathfrak m ) $ , where $ h $ is the order of the absolute class group and $ \phi $ is Euler's phi-function. For different conductors $ \mathfrak m _{1} $ and $ \mathfrak m _{2} $ the class groups may be equivalent if $ H _ {\mathfrak m _{1}} $ and $ H _ {\mathfrak m _{2}} $ consists in fact of the same progressions $ ( \mathop{\rm mod}\nolimits \ f \ ) $ , where $ f = \mathop{\rm l}\nolimits.c.m. ( \mathfrak m _{1} ,\ \mathfrak m _{2} ) $ . If one agrees not to distinguish between equivalent class groups, then one obtains the concept of a class group according to Weber with a conductor $ f $ that is the greatest common divisor of all equivalent conductors. A class field according to Weber is a field $ K / k $ in which only the prime ideals in its principal class $ H _{f} $ , and only these, split completely. Dirichlet's theorem on prime ideals in progressions, which is valid in every field $ k $ , implies that there are infinitely many prime ideals. Weber has shown that in certain particular cases the Galois group $ \mathop{\rm Gal}\nolimits ( K / k ) $ of a class field and the class group $ A _{f} / H _{f} $ of $ k $ are isomorphic.

A new point of view on class field theory, which is still valid, originated with D. Hilbert. He understood that between all relatively Abelian extensions of a field $ k $ and all class fields for this field there must exist a one-to-one correspondence. This correspondence can be stated as follows. If for some conductor $ f $ one constructs the Weber class group, then there is only one normal extension $ K / k $ in which the prime ideals of Weber's principal class, and only these, will split completely. Moreover, the Galois group of $ K / k $ is isomorphic to the Weber class group and the discriminant $ D $ of $ K / k $ consists of the same prime ideals as the conductor $ f $ . The converse is also true: If an Abelian extension with Galois group $ \mathop{\rm Gal}\nolimits ( K / k ) $ is given, then there exists a method (subsequently explicitly formulated by Takagi) by which one can uniquely construct a principal class $ H _{f} $ such that the class group $ A _{f} / H _{f} $ is isomorphic to $ \mathop{\rm Gal}\nolimits ( K / k ) $ , only the prime ideals of $ H _{f} $ split completely in $ K $ , and such that the conductor $ f $ has the same prime divisors as the discriminant $ D $ of $ K / k $ . The "duality" was stated by Hilbert in 1900 as a hypothesis (he proved it only in special cases). In its general form it was proved by Takagi.

The next important stage of development of class group theory is related to the name of E. Artin, who revealed the special role of the canonical isomorphism between the Galois group and the ideal class group. He proved that this isomorphism is given by the Frobenius automorphism $ \sigma _{\mathfrak p} $ of an Abelian extension $ K / k $ , defined as $$ a ^ {N _{\mathfrak p}} \equiv \sigma _ {\mathfrak p} a ( \mathop{\rm mod}\nolimits \ \mathfrak p ) . $$ Here $ N _{\mathfrak p} $ is the absolute norm of the ideal $ \mathfrak p $ , $ a $ runs through all numbers of $ K $ and $ \mathfrak p $ is a prime ideal of $ k $ . The automorphism $ \sigma _{\mathfrak p} $ (nowadays denoted by $$ \left . \left ( \frac{K / k}{\mathfrak p} \right ) \ \right ) $$ depends (in an appropriate group) only on the ideal class to which $ \mathfrak p $ belongs. It is multiplicative: $$ \left ( \frac{K / k}{\mathfrak p _{1}} \right ) \left ( \frac{K / k}{\mathfrak p _{2}} \right ) = \left ( \frac{K / k}{\mathfrak p _{1} \mathfrak p _{2}} \right ) , $$ where the symbol at the right-hand side is understood as an automorphism of the class to which $ \mathfrak p _{1} \mathfrak p _{2} $ belongs. With this in mind, Artin introduced the symbol $$ \left ( \frac{K / k}{\mathfrak a} \right ) $$ on the entire group $ A _{f} $ of ideals $ \mathfrak a $ of $ k $ that are relatively prime with the conductor $ f $ . It is called the Artin symbol and realizes an isomorphism between the class group $ A _{f} / H _{f} $ and the Galois group $ \mathop{\rm Gal}\nolimits ( K / k ) $ , whose explicit from is expressed in Artin's reciprocity law: $$ \left ( \frac{K / k}{\mathfrak a} \right ) = 1 $$ if and only if $ \mathfrak a \in H _{f} $ (reciprocity as a correspondence between the groups $ \mathop{\rm Gal}\nolimits ( K / k ) $ and $ A _{f} / H _{f} $ ). From this one can obtain the classical form of the reciprocity law in the language of Kummer's power reciprocity symbol (one has to consider the field $ K = k ( a ^{1}/n ) $ , cf. Reciprocity laws). This form, in turn, implies the reciprocity law for Hilbert's symbol. In all three forms the reciprocity law is regarded as an isomorphism of groups, and the symbols of Artin, Kummer and Hilbert are regarded as group elements realizing this isomorphism. However, each of them also has a numerical value, which is equal to some $ n $ -th root of unity. Therefore one can formulate the reciprocity law as follows. Given the value of $ ( \alpha / \beta ) _{n} $ , it is required to find the value of the symbol $ ( \alpha / \beta ) _{n} $ reciprocal to it, i.e. to exhibit the explicit form of the function of $ ( \alpha ,\ \beta ) $ defined by $$ \left ( \frac \alpha \beta \right ) \left ( \frac \beta \alpha \right ) ^{-1} . $$ In this form the law first appeared in the work of C.F. Gauss (cf. Gauss reciprocity law) for quadratic fields and in the work of Kummer for cyclotomic fields of prime degree. The study of the reciprocity law in this form was subsequently continued by C.G.J. Jacobi, F.G.M. Eisenstein, Hilbert, H. Hasse, and others, but they only obtained partial results. I.R. Shafarevich [6] solved this problem in its general form in 1948, basing himself on the idea of establishing a connection between the arithmetic definition of the symbol $$ \left ( \frac{\alpha ,\ \beta}{\mathfrak p} \right ) $$ and the analytic definition of an Abelian differential $ \alpha \cdot d \beta $ on a Riemann surface. He gave a construction of $$ \left ( \frac{\alpha ,\ \beta}{\mathfrak p} \right ) $$ which precisely corresponds to the determination of the residue of $ \alpha \cdot d \beta $ in the $ \mathfrak p $ -adic field. For this purpose he introduced $ E $ -functions, also called Shafarevich functions, in terms of which he also obtained an explicit form of the reciprocity law in the general case.

In the late 1920s Hasse introduced the idea of doing class field theory for one prime ideal of $ k $ at a time, and reformulated and proved many theorems for an Abelian extension $ K _{p} $ of a local field $ k _{p} $ . At first the idea was of secondary importance only (the results were obtained as a consequence of the local theory), but at the end of the 1930s C. Chevalley proved the extraordinary importance of the local point of view in the creation of class field theory. He introduced the concept of an idèle group and formulated general class field theory from the local point of view. Following this the local-global principle became established in class field theory. Subsequently it was extended and refined (cf. [5]), and as a result Abelian class field theory took a structured and completed form. The problem of creating non-Abelian class field theory for normal extensions with non-Abelian Galois group remains.

The exposition above relates mainly to the qualitative aspects of algebraic number theory. In questions of quantitative estimation and methods algebraic number theory is intimately connected with analytic number theory. It is also based, to a large extent, on properties of $ \zeta $ -functions and $ L $ -functions of algebraic fields.

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Comments[edit]

An algebraic number field of degree $ n $ is a degree $ n $ extension of the field $ \mathbf Q $ of rational numbers. An element $ \alpha \in K $ is an algebraic integer if it is integral over $ \mathbf Z $ , i.e. if it is a root of a monic polynomial in $ \mathbf Z [ X ] $ . These form a free Abelian subgroup of rank $ n $ , called the ring of integers $ O _{K} $ of $ K $ and also the principal order of $ K $ ; a basis of $ O _{K} $ is an integral basis, also called a minimal basis.

For Kummer's approach to the Fermat problem cf. [a3]. The link (equivalence) between Kummer's ideal numbers and ideals comes from the fact that for the prime ideals in $ O _{K} $ there is a field extension $ L / K $ in which they become principal ideals.

The relative degree $ f _{i} $ of an ideal $ \mathfrak P _{i} $ is the degree of the residue field extension $ O _{L} / \mathfrak P _{i} $ over $ O _{K} / \mathfrak p $ where $ \mathfrak p = \mathfrak P _{i} \cap O _{K} $ .

For more details, definitions, newer results, and more precise statements of much of the above cf. various more specified articles in this Encyclopaedia. In particular the articles Algebraic number; Unit; Extension of a field; Regulator of an algebraic number field; Discriminant; Frobenius automorphism (also for Artin symbol), Kummer theorem on decomposition of ideals, and especially class field theory.

The ideal class groups $ A _{\mathfrak m} / H _{\mathfrak m} $ are also often known as ray class groups $ \mathop{\rm mod}\nolimits \ \mathfrak m $ , and the corresponding fields as ray class fields. The class field corresponding to the ray class group $ A _{1} / L _{1} $ is the Hilbert class field. The definitions above ignore some problems with the Archimedean primes; in particular the principal ideals $ ( \alpha ) $ making up $ L _{\mathfrak m} $ must have $ \alpha $ totally positive, i.e. such that all real conjugates of $ \alpha $ are $ > 0 $ .

The discussion of class field theory above (up to the mention of idèles) is in terms of the older ideal-theoretic formulation as in e.g. Hasse's Zahlbericht [a1], which in fact can be advantageously consulted for more details and precise statements. For a discussion of the relations between the ideal-theoretic formulations and the modern idèle based class field theory [a2] is recommended.

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