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Algebraic number field
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In mathematics, an algebraic number field (or simply number field) F is a finite, (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension, when considered as a vector space over Q.
The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
The regular representation, trace and determinant
Suppose F is a field extension of the field of rational numbers Q of finite degree n.
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Encyclopedia
In mathematics, an algebraic number field (or simply number field) F is a finite, (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension, when considered as a vector space over Q.
The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
The regular representation, trace and determinant
Suppose F is a field extension of the field of rational numbers Q of finite degree n. This means that F is an n-dimensional vector space over Q, elements of F form a commutative ring (in fact a field) under the operations of addition and multiplication, and all non-zero elements of F are invertible.
Let us choose a basis e1, ..., en for F, then any element x of F has a unique representation in the form x = ? xi ei. Using the multiplication in F, we may represent the elements of the field F by n by n matrices, as follows:
This way of associating a matrix to any element of the field F is called the regular representation. The square matrix A = A(x) with the rational entries aij, where i and j are indices between 1 and n, represents the effect of multiplication by x in the basis e. It follows that if the element y of F is represented by a matrix B, then the product xy is represented by the matrix product AB. Invariants of matrices, such as the trace, determinant, and characteristic polynomial, depend solely on the field element x and not on the basis. In particular, the trace of the matrix A(x) is called the trace of the field element x and denoted Tr(x), and the determinant is called the norm of x and denoted N(x).
Properties
Let ? be a rational number, or as it is common to say, a scalar, and x, y be two elements of F, then the trace and determinant have the following properties:
- Tr(x + y) = Tr(x) + Tr(y)
- Tr(?x) = ? Tr(x)
- N(xy) = N(x) N(y)
- N(?x) = ?n N(x)
The first two properties express the fact that the trace is a linear function of x. The third property is the multiplicativity of the norm, and the last property means that the norm is a homogeneous function of x of degree n.
Algebraic integers
An element x of the algebraic number field F is called an algebraic integer
if it is a root of a monic polynomial with integer coefficients. Algebraic integers admit other, equivalent descriptions. An element x of F is an algebraic integer if and only if the characteristic polynomial pA of the matrix A associated to x is a monic polynomial with integer coefficients. Suppose that the matrix A that represents an element x has integer entries in some basis e. By the Cayley-Hamilton theorem, pA(A) = 0, and it follows that pA(x) = 0, so that x is an algebraic integer. Conversely, if x is an element of F which is a root of a monic polynomial with integer coefficients then the same property holds for the corresponding matrix A. In this case it can be proven that A is an integer matrix in a suitable basis of F. Note that the property of being an algebraic integer is defined in a way that is independent of a choice of a basis in F.
The set of integral square matrices is closed under addition and multiplication, and it follows that the algebraic integers in F form a ring, denoted by OF, which is a subring of F. A field contains no zero divisors and this property is inherited by any subring. Therefore, the ring of integers of F is an integral domain. The field F is the field of fractions of the integral domain OF.
Properties
An abstract commutative ring with these three properties is called a Dedekind ring (or Dedekind domain), in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers.
Bases for number fields
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 proper subfield, hence generates F and has an irreducible minimal 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.
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.
Integral basis
An integral basis for a number field F of degree n is a set B = 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 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 to have built-in programs to do this.
Trace form and discriminant
We may define a bilinear form on F by means of the trace, by T(x y); 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.
Example
Consider F = Q(x), where x satisfies x3 − 11x2 + x + 1 = 0. Then an integral basis is [1, x, 1/2(x2 + 1)], and the corresponding integral trace form is
The "3" in the upper left hand corner of this matrix is the trace of the matrix of the map defined by the first basis element (1) in the regular representation of F on F. This basis element induces the identity map on the 3-dimensional vector space, F. The trace of the matrix of the identity map on a 3-dimensional vector space is 3.
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.
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.
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.
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.
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 |?|t = |Nt(?)|p1/m. Such an absolute value is called an ultrametric, non-Archimedean or p-adic place of F.
An example
For an example, consider the factorization of the polynomial
over the 23-adic numbers Q23. Up to 529 = 232 this factorization is
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 suffices. 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 . 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.)
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.
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.
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.
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 relate the two by setting
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.
See also
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