In algebraic number theory

Algebraic number theory

Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...

, the narrow class group of a number field K is a refinement of the class group of K that takes into account some information about embeddings of K into the field of real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s.

Formal definition

Suppose that K is a finite extension of Q. Recall that the ordinary class group of K is defined to be
where I_K is the group of fractional ideal

Fractional ideal

In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed...

s of K, and P_K is the group of principal fractional ideals of K, that is, ideals of the form aO_K where a is a unit

Unit (ring theory)

In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...

 of K.

The narrow class group is defined to be the quotient
where now P_K⁺ is the group of totally positive principal fractional ideals of K; that is, ideals of the form aO_K where a is a unit of K such that σ(a) is positive for every embedding

Uses

The narrow class group features prominently in the theory of representing of integers by quadratic form

Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

s. An example is the following result (Fröhlich and Taylor, Chapter V, Theorem 1.25).

Theorem. Suppose that

where d is a square-free integer

Square-free integer

In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...

, and that the narrow class group of K is trivial. Suppose that

is a basis for the ring of integers of K. Define a quadratic form

,

where N_K/Q is the norm

Field norm

In mathematics, the norm is a mapping defined in field theory, to map elements of a larger field into a smaller one.-Formal definitions:1. Let K be a field and L a finite extension of K...

. Then a prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 p is of the form

for some integers x and y if and only if either

or

or

where d_K is the discriminant of K, and

indicates the Legendre symbol

Legendre symbol

In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number p: its value on a quadratic residue mod p is 1 and on a quadratic non-residue is −1....

.

Examples

For example, one can prove that the quadratic fields Q(√−1), Q(√2), Q(√−3) all have trivial narrow class group. Then, by choosing appropriate bases for the integers of each of these fields, the above theorem implies the following:

• A prime p is of the form p = x^{2 + y2 for integers x and y if and only if}

(This is known as Fermat's theorem on sums of two squares.)

• A prime p is of the form p = x^{2 − 2y2 for integers x and y if and only if}

• A prime p is of the form p = x^{2 − xy + y2 for integers x and y if and only if}

(cf. Eisenstein prime

Eisenstein prime

In mathematics, an Eisenstein prime is an Eisenstein integerz = a + b\,\omega\qquadthat is irreducible in the ring-theoretic sense: its only Eisenstein divisors are the units , a + bω itself and its associates.The associates and the complex conjugate...

)