canonical form of element of number field

TheoremMathworldPlanetmath.  Let ϑ be an algebraic numberMathworldPlanetmath of degree ( n.  Any element α of the algebraic number fieldMathworldPlanetmath (ϑ) may be uniquely expressed in the canonical form

α=c0+c1ϑ+c2ϑ2++cn-1ϑn-1 (1)

where the numbers ci are rational.

Proof.  We start from the fact that (ϑ) consists of all expressions formed of ϑ and rational numbers using arithmetic operations (no divisorMathworldPlanetmathPlanetmath ( must vanish); such expressions lead always to the form

α=a(ϑ)b(ϑ) (2)

where the numerator and the denominator are polynomialsMathworldPlanetmathPlanetmathPlanetmath in ϑ with rational coefficients (which can, in fact, be chosen integers).

So, let α in (2) an arbitrary element of the field (ϑ).  Denote by f(x) the minimal polynomial of ϑ over .  Since  b(ϑ)0,  the polynomial f(x) does not divide ( b(x), and since f(x) is irreduciblePlanetmathPlanetmath (, the greatest common divisorMathworldPlanetmathPlanetmath ( of f(x) and b(x) is a constant polynomial, which can be normed to 1.  Thus there exist the polynomials φ(x) and ψ(x) of the ring [x] such that

φ(x)f(x)+ψ(x)b(x) 1.


φ(ϑ)f(ϑ)= 0+ψ(ϑ)b(ϑ)= 1,



and consequently


Hence, α is a polynomial in ϑ with rational coefficients.

Let now

ψ1(x)=q(x)f(x)+r(x)  with deg(r)<deg(f)=n.



It follows that


whence (1) is true.

Suppose that we had also


with every di rational.  This implies that

(cn-1-dn-1)ϑn-1++(c1-d1)ϑ+(c0-d0)= 0,

i.e. that ϑ satisfies the equation

(cn-1-dn-1)xn-1++(c1-d1)x+(c0-d0)= 0

with rational coefficients and degree less than n.  Because the degree of ϑ is n, it is possible only if all differencesPlanetmathPlanetmath ci-di vanish.  Thus


i.e. the (1) is unique.

Note 1.  The polynomial c0+c1x++cn-1xn-1 is called the canonical polynomial of the algebraic number α with respect to the primitive elementMathworldPlanetmathPlanetmath ( ϑ.

Note 2.  The theorem allows to denote the field (ϑ) similarly as polynomial rings: [ϑ].

Note 3.  When allowed, unlike in (1), higher powers of the primitive element ϑ (whose minimal polynomial is xn+a1xn-1++an), one may unlimitedly write different sum of α, e.g.

α =(c0+c1ϑ++cn-1ϑn-1)+(ϑn+a1ϑn-1++an)
Title canonical form of element of number field
Canonical name CanonicalFormOfElementOfNumberField
Date of creation 2013-03-22 19:08:00
Last modified on 2013-03-22 19:08:00
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Theorem
Classification msc 11R04
Related topic Canonical
Related topic SimpleFieldExtension
Related topic CanonicalBasis
Related topic IntegralBasis
Defines canonical form
Defines canonical form in number field
Defines canonical polynomial