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In mathematics, an algebraic number field (or simply number field) is a finite-dimensional (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension, or degree, when considered as a vector space over Q. For other meanings of mathematics or math, see mathematics (disambiguation). ...
In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...
In abstract algebra, a field extension L /K is called algebraic if every element of L is algebraic over K, i. ...
In abstract algebra, a subfield of a field L is a subset K of L which is closed under the addition and multiplication operations of L and itself forms a field with these operations. ...
In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, the dimension of a vector space V is the cardinality (i. ...
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the size of the extension. ...
In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ...
The study of algebraic number fields, and these days also of infinite algebraic extensions of the rational number field, is the central topic of algebraic number theory. Algebraic number theory is a branch of number theory in which the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. ...
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The regular representation
Suppose F is a field extension of finite dimension n of the rational numbers Q. This means F is an n-dimensional vector space over Q with the structure of a field. Hence, if v1, ..., vn is a basis for F, then for any element x of F we may write This means that multiplication by x may be represented by a square matrix X = [aij] with rational elements; this is the regular representation of x for the basis given by vi. Invariants of the matrix, such as the trace, determinant, and characteristic polynomial, are properties of the field element x and do not depend on the basis. In mathematics, an invariant is something that does not change under a set of transformations. ...
It has been suggested that this article or section be merged with Tracing. ...
In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ...
The characteristic polynomial xn + c1xn − 1 + ... + cn of X gives a monic polynomial with coefficients in Q satisfied by x. The trace of X is minus the c1 coefficient of this polynomial, and is a function T(x) depending only on x. The determinant of X is (−1)ncn, and defines another function depending only on x, N(x), called the norm of x. Let a be in Q and x,y be in F, then we have the following properties: In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ...
In mathematics, a monic can refer to monic morphism – a special kind of morphism in category theory, monic polynomial – a polynomial whose leading coefficient is one. ...
In mathematics, the (field) norm is a mapping defined in field theory, to map elements of a larger field into a smaller one. ...
- T(x + y) = T(x) + T(y)
- T(ax) = a T(x)
- N(xy) = N(x)N(y)
- N(ax) = an N(x)
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Algebraic integers If the matrix X representing the field element x has a characteristic polynomial whose coefficients are integers, the element x is called an algebraic integer; since the characteristic polynomial of X depends only on x this defines a property of x. The matrix X in this case is similar to an integral matrix (a matrix with only integer coefficients) X'; and we may use a basis for F consisting of algebraic integers, called an integral basis, which is always possible, in which every algebraic integer x corresponds to an integral matrix X. Square integral matrices form a ring; the sums and products of integral matrices are integral matrices, and so the square integral matrices representing algebraic integers of F form a subring. Because the elements of this subring represent elements of a field, the subring contains no zero divisors. Hence, the ring of integers of F, which can be represented by means of this subring, is an integral domain. This ring is an inherent property of the field F, and is often denoted OF. The field F in turn is the field of fractions of the integral domain OF. In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ...
In mathematics, the phrase ring of integers can refer to the ring of all integers (whole numbers: positive, negative or zero), usually denoted by Z. given an algebraic number field K, the algebraic integers contained in K form a ring, the ring of integers of K, often denoted by OK...
In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...
In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ...
In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ...
In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions of the integral domain. ...
Since OF consists of all of the integers of F, it is integrally closed in its field of fractions. It is also a Noetherian ring, and every nonzero prime ideal is a maximal ideal. Such a ring is called a Dedekind ring (or Dedekind domain), and the rings of integers of algebraic number fields are the classic examples of Dedekind rings. In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ...
In abstract algebra, a Dedekind domain, named after Richard Dedekind, is a Noetherian integral domain which is integrally closed in its fraction field and which has Krull dimension 1. ...
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Bases for number fields [edit]
Power basis Since there are only a finite number of subfields of F, and since these correspond to subspaces of F as a vector space over Q, in general an element of F does not belong to any subfield, and hence has an irreducible characteristic polynomial over Q. Such an element x is called a primitive element, and the primitive element theorem tells us that extensions of fields of characteristic zero indeed have a primitive element. In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ...
In mathematics, a primitive element for an extension of fields L/K is an element ζ of L such that L = K(ζ), or in other words such that L is generated by ζ over K. This means that every element of L can be written as a quotient of two polynomials in...
In mathematics, a primitive element for an extension of fields L/K is an element ζ of L such that L = K(ζ), or in other words such that L is generated by ζ over K. This means that every element of L can be written as a quotient of...
In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...
If x is a primitive element, then [1, x, x2, ..., xn − 1] is a basis for F. If the characteristic polynomial for x has non-integral coefficients, then we may find the greatest common divisor D of the denominators of the coefficients, and take instead the polynomial for y = Dx which we may obtain by substituting y/D for x in the polynomial for x. This gives us an integral power basis, defined in terms of a single root of an irreducible monic polynomial of degree n over Q with integer coefficients. In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers. ...
In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...
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Integral basis An integral basis for a number field F of degree n is a set B = {b1,...,bn} of n algebraic integers in F such that every element of the ring of integers OF of F can be written uniquely as a Z-linear combination of elements of B; that is, for any x in OF in we have x = m1b1+...+mnbn, where the mi are (ordinary) integers. It is then also the case that any element of F can be written uniquely as m1b1+...+mnbn, where now the mi are rational numbers. The algebraic integers of F are then precisely those elements of F where the mi are all integers.
Working locally and using tools such as the Frobenius map, it is always possible to explicitly compute such a basis, and it is now standard for computer algebra systems such as Maple and Mathematica to have built-in programs to do this. In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ...
In commutative algebra and field theory, which are branches of mathematics, the Frobenius endomorphism is a special endomorphism of rings with prime characteristic, a class importantly including fields. ...
A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ...
Maple 9. ...
This article is about computer software. ...
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Trace form and discriminant We may define a bilinear form on F by means of the trace, by T(xy); this is called the trace form. If b1, ..., bn is an integral basis for F, then we may define a symmetric integral matrix, the integral trace form, by tij = T(bibj). Then the discriminant of F may be defined as det(t). It is an integer, and is an invariant property of the field F, not depending on the choice of integral basis. In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ...
In mathematics, the discriminant of an algebraic number field is a numerical invariant containing information about ramified primes. ...
For example, suppose F = Q(x), where x satisifies x3 − 11x2 + x + 1 = 0. Then an integral basis is [1, x, (x2 + 1)/2], and the corresponding integral trace form is
The determinant of this is 1304 = 23 163, the field discriminant; in comparison the root discriminant, or discriminant of the polynomial, is 5216 = 25 163. In mathematics, a discriminant is an expression which discriminates qualities of algebraic structures. ...
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Places Mathematicians of the nineteenth century assumed that algebraic numbers were a type of complex number. This situation changed with the discovery of p-adic numbers by Hensel in 1897; and now it is standard to consider all of the various possible embeddings of a number field F into its various topological completions at once. The p-adic number systems were first described by Kurt Hensel in 1897. ...
Abigail and Brittany Hensel (Hensel twins) (-) Kurt Hensel (1861-1941), mathematician Walther Hensel (1887-1956) Hensel Family Wilhelm Hensel (1794-1861), painter = (married) Fanny Mendelssohn Hensel (-), sister of Felix Mendelssohn Paul Hensel (1860-1930) Luise Hensel (1798-1876), poet Hensels lemma See also Hensell Hänsel(Haensel), Hänsell...
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Archimedean places Given an irreducible polynomial f over Q defining a primitive element x of a number field F, and hence a power basis for F, we may factor f into irreducible factors over the real numbers R. These factors are either of degree one or two, and since there are no repeated roots, there are no repeated factors. Each factor of degree one gives a real root, and by replacing x by the real root r, we obtain an embedding into the real numbers; the number of such embeddings is equal to the number of real roots. This allows us to define an absolute value on the elements of F, since they are now elements of R; such an absolute value is called a real place of the number field F. Similarly, for each factor of degree two we obtain a pair of conjugate complex numbers, which allows for two conjugate embeddings into C. Either one of this pair of embeddings can be used to define an absolute value on F, which is the same for both embeddings since they are conjugate. This absolute value is called a complex place of F. These are the Archimedean places of F, corresponding to Archimedean absolute values. In mathematics, an absolute value is a function which measures the size of elements in a field or integral domain. ...
In mathematics, an absolute value is a function which measures the size of elements in a field or integral domain. ...
Archimedean refers to a thing named after the Greek mathematician Archimedes. ...
In mathematics, the absolute value (or modulus1) of a real number is its distance from zero on a number line. ...
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Ultrametric places The real numbers are a topological completion of the rational numbers, but not the only one. Given the usual absolute value, we can define a Cauchy sequence in terms of |xn − xm|, and a null sequence as a sequence with absolute value tending towards zero. Null sequences are a maximal ideal in the ring of Cauchy sequences, and by taking the quotient ring we obtain a field, the field of real numbers. By Ostrowski's theorem, the non-trivial absolute values on Q are, up to equivalence, the usual real absolute value, and the p-adic absolute values defined for each prime number p. Given a prime p, we may define the p-adic absolute value on rational numbers q = pn a/b, where a and b are integers not divisible by p, as |q|p = p−n. We may now define p-adic Cauchy sequences and null sequences in terms of this absolute value, and by taking the quotient ring obtain another completion of the rational numbers, the p-adic numbers. In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ...
In mathematics, more specifically in ring theory a maximal ideal is a special kind of ideal which is in some sense maximal, that is not contained in any other non-trivial ideal of the ring. ...
Ostrowskis theorem, due to Alexander Ostrowski, states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value. ...
The p-adic number systems were first described by Kurt Hensel in 1897. ...
In mathematics, an absolute value is a function which measures the size of elements in a field or integral domain. ...
Factoring the polynomial f of degree n satisfied by the primitive element x, we now may obtain factors of various degrees, none of which are repeated, and the degrees of which add up to n. For each of these p-adically irreducible factors t, we may suppose that x satisfies t and obtain an embedding of F into an algebraic extension of finite degree over Qp. Such a local field behaves in many ways like a number field, and the p-adic numbers may similarly play the role of the rationals; in particular, we can define the norm and trace in exactly the same way, now giving functions mapping to Qp. By using this p-adic norm map Nt for the place t, we may define an absolute value corresponding to a given p-adically irreducible factor t of degree m by |θ|t = |Nt(θ)|p1/m. Such an absolute value is called an ultrametric, non-Archimedean or p-adic place of F. In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. ...
In mathematics, an ultrametric space is a special kind of metric space. ...
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An example For an example, consider the factorization of the polynomial f = x3 − x − 1 over the 23-adic numbers Q23. Up to 529 = 232 this factorization is f = (x + 181)(x2 − 181x − 38) = f1f2. While this corresponds to less than three digits of accuracy, the factorization is easily lifted to much more accurate ones involving higher powers of 23, and in any case already sufficies. If we consider the element y = x − 10 of Q23, then by substituting x = y + 10 into the first factor f1 modulo 529, we obtain y + 191, so the valuation |y|f1 for y given by f1 is |−191|23 = 1. On the other hand if we substitute x = y + 10 into f2, we obtain y2 − 161y − 161 modulo 529. Since 161 = 7×23, we find that Since possible values for the absolute value of the place defined by the factor f2 are not confined to integer powers of 23, but instead are integer powers of the square root of 23, the place is said to be ramified with ramification index two.
The valuations of any element of F can be computed in this way using resultants. If, for example y = x2 − x − 1, using the resultant to eliminate x between this relationship and f = x3 − x − 1 = 0 gives y3 − 5y2 + 4y − 1 = 0. If instead we eliminate with respect to the factors f1 and f2 of f, we obtain the corresponding factors for the polynomial for y, and then the 23-adic valuation applied to the constant (norm) term allows us to compute the valuations of y for f1 and f2 (which are both 1 in this instance.) In mathematics, the resultant of two monic polynomials and over a field is defined as the product of the differences of their roots, where and take on values in the algebraic closure of . ...
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Dedekind discriminant theorem Much of the significance of the discriminant lies in the fact that ramified ultrametric places are all places obtained from factorizations in Qp 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 disciminant is needed. This is the Dedekind discriminant theorem. In the example above, the discriminant of the number field Q(x) with x3 − x − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. 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. In mathematics, an algebraic number field (or simply number field) is a finite-dimensional (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension, or degree, when considered as a vector space over Q. The study of...
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Prime ideals For any ultrametric place t we have that |x|t ≤ 1 for any x in OF, since the minimal polynomial for x has integer factors, and hence its p-adic factorization has factors in Zp. Consequently, the norm term (constant term) for each factor is a p-adic integer, and one of these is the integer used for defining the absolute value for t.
If we take the subset of OF defined by |x|t < 1, then we obtain an ideal P of OF. This is because by the ultrametric propery the sum of any two elements of P is in P, and if x is in OF and y is in P, then |xy|t = |x|t|y|t < 1. If |xy|t < 1 with both |x|t ≤ 1 and |x|t ≤ 1, then at least one of x and y must be in P. Hence, P is a prime ideal of OF. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...
In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ...
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Localization Given an ultrametric place t on a number field F, the corresponding local ring, or localization, is the subring T of F of all elements x such that |x|t ≤ 1. By the ultrametric propery T is a ring, and since every integer x of F satisfies |x|t ≤ 1, OF is contained in T. For every element x of F, at least one of x or x−1 is contained in T. Hence T is a valuation ring. In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ...
Localization can mean any of the following: Generally, localization is the determination of the locality (position) of an object. ...
In abstract algebra, local rings are certain rings that are comparatively simple and serve to describe the local behavior of functions defined on varieties or manifolds. ...
The valuation group of T, F*/T*, is isomorphic to the integers, and so T is a discrete valuation ring. The place t is defined p-adically for some p, and is said to "lie over" p. The mapping ν to the integers by the valuation map maps p to some positive integer ν(p) = e, which is the ramification index. Since |p|t = 1/p, we can related the two by setting In mathematics, a discrete valuation ring (DVR) is a particular kind of commutative ring that is a local ring, which satisfies conditions that in algebraic geometry come from non-singularity of a point on an algebraic curve. ...
Given a prime ideal P, we can also construct the localization of F at P by taking all ratios a/b such that a is any element of OF and b is any element of OF which does not belong to P. Hence we can define a three-way equivalency between ultrametric absolute values, prime ideals, and localizations on a number field, and starting from any of them we can construct the other two. [edit]
See also [edit] In mathematics, a quadratic field is a field extension K/Q of the form where d is a non-zero rational number. ...
In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ...
In mathematics, the discriminant of an algebraic number field is a numerical invariant containing information about ramified primes. ...
In mathematics the theory of algebraic number fields gives rise to a finite abelian group constructed from each such field, its ideal class group. ...
In algebraic number theory, Dirichlets unit theorem determines the rank of the group of units in the ring OK of algebraic integers of a number field K. The statement is that the rank is r + s − 1 where r is the number of real embeddings and 2s the number...
In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. ...
The term global field refers to either of the following: a number field, i. ...
In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian. ...
In mathematics, a Kummer extension of fields is a field extension L/K where for some given integer n > 1 we have [L:K] = n and L is generated over K by a root of a polynomial Xn − a with a in K, and K contains n distinct roots...
In mathematics, reciprocity is applied to a number of theorems, and at times certain relationships. ...
In mathematics, class field theory is a major branch of algebraic number theory. ...
In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. It is an abelian group with elements isomorphism classes of division algebras over K, such that the center is exactly K. The group is named for the algebraist Richard Brauer. ...
In number theory, Iwasawa theory is a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa as part of the theory of cyclotomic fields. ...
In mathematics, the Dedekind zeta-function is a Dirichlet series defined for any algebraic number field , and denoted where is a complex variable. ...
References - Gerald J. Janusz, Algebraic Number Fields, second edition, American Mathematical Society, 1995
- Serge Lang, Algebraic Number Theory, second edition, Springer, 2000
- Richard A. Mollin, Algebraic Number Theory, CRC, 1999
- Andre Weil, Basic Number Theory, third edition, Springer, 1995
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