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.

Contents 1 The regular representation, trace and determinant 1.1 Properties 2 Algebraic integers 2.1 Properties 3 Bases for number fields 3.1 Power basis 3.2 Integral basis 4 Trace form and discriminant 4.1 Example 5 Places 5.1 Archimedean places 5.2 Ultrametric places 5.3 An example 5.4 Dedekind discriminant theorem 5.5 Prime ideals 5.6 Localization 6 See also 7 References

[edit] 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 under the operations of addition and multiplication, and all non-zero elements of F are invertible. Let us choose a basis e₁, ..., e_n for F, then any element x of F has a unique representation in the form x = ∑ x_i e_i. 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 a_ij, 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).

[edit] Properties

Let λ 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(λx) = λ Tr(x)
• N(xy) = N(x) N(y)
• N(λx) = λⁿ 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.

[edit] 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 p_A 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, p_A(A) = 0, and it follows that p_A(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 O_F, 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 O_F.

[edit] Properties

• The ring of algebraic integers O_F is an integral domain that is integrally closed in its field of fractions F.
• It is a Noetherian ring.
• Every nonzero prime ideal of O_F is maximal.

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.

[edit] 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 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, ..., x^n − 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.

[edit] Integral basis

An integral basis for a number field F of degree n is a set B = {b₁, …, b_n} of n algebraic integers in F such that every element of the ring of integers O_F of F can be written uniquely as a Z-linear combination of elements of B; that is, for any x in O_F we have x = m₁b₁ + … + m_nb_n, where the m_i are (ordinary) integers. It is then also the case that any element of F can be written uniquely as m₁b₁ + … + m_nb_n, where now the m_i are rational numbers. The algebraic integers of F are then precisely those elements of F where the m_i 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.

[edit] 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, ..., b_n is an integral basis for F, then we may define a symmetric integral matrix, the integral trace form, by t_ij = T(b_ib_j). 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.

[edit] 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 determinant of this is 1304 = 2³ 163, the field discriminant; in comparison the root discriminant, or discriminant of the polynomial, is 5216 = 2⁵ 163.

[edit] 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.

[edit] 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.

[edit] 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 |x_n − x_m|, 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 Q_p. 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 Q_p. By using this p-adic norm map N_t 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 = |N_t(θ)|_p^1/m. Such an absolute value is called an ultrametric, non-Archimedean or p-adic place of F.

[edit] 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 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.)

[edit] 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 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 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.

[edit] Prime ideals

For any ultrametric place t we have that |x|_t ≤ 1 for any x in O_F, since the minimal polynomial for x has integer factors, and hence its p-adic factorization has factors in Z_p. 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 O_F defined by |x|_t < 1, then we obtain an ideal P of O_F. This is because by the ultrametric propery the sum of any two elements of P is in P, and if x is in O_F 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 O_F.

[edit] 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, O_F 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 O_F and b is any element of O_F 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] 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