Algebraic Number Field - Algebraicity and Ring of Integers

Algebraicity and Ring of Integers

Generally, in abstract algebra, a field extension F / E is algebraic if every element f of the bigger field F is the zero of a polynomial with coefficients e₀, ..., e_m in E:

p(f) = e_mfm + e_m−1fm−1 + ... + e₁f + e₀ = 0.

It is a fact that every finite field extension is algebraic (proof: for x in F simply consider x, x^2, x^3 ...we get a linear dependence, i.e. a polynomial x is a root of!). In particular this applies to algebraic number fields, so any element f of an algebraic number field F can be written as a zero of a polynomial with rational coefficients. Therefore, elements of F are also referred to as algebraic numbers. Given a polynomial p such that p(f) = 0, it can be arranged such that the leading coefficient e_m is one, by dividing all coefficients by it, if necessary. A polynomial with this property is known as a monic polynomial. In general it will have rational coefficients. If, however, its coefficients are actually all integers, f is called an algebraic integer. Any (usual) integer z ∈ Z is an algebraic integer, as it is the zero of the linear monic polynomial:

p(t) = t − z.

It can be shown that any algebraic integer that is also a rational number must actually be an integer, whence the name "algebraic integer". Again using abstract algebra, specifically the notion of a finitely generated module, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer, it follows that the algebraic integers in F form a ring denoted O_F called the ring of integers of F. It is a subring of (that is, a ring contained in) 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. This way one can get back and forth between the algebraic number field F and its ring of integers O_F. Rings of algebraic integers have three distinctive properties: firstly, O_F is an integral domain that is integrally closed in its field of fractions F. Secondly, O_F is a Noetherian ring. Finally, every nonzero prime ideal of O_F is maximal or, equivalently, the Krull dimension of this ring is one. 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.