canonical form of element of number field
where the numbers are rational.
Proof. We start from the fact that consists of all expressions formed of and rational numbers using arithmetic operations (no divisor (http://planetmath.org/Division) must vanish); such expressions lead always to the form
So, let in (2) an arbitrary element of the field . Denote by the minimal polynomial of over . Since , the polynomial does not divide (http://planetmath.org/DivisibilityInRings) , and since is irreducible (http://planetmath.org/IrreduciblePolynomial2), the greatest common divisor (http://planetmath.org/PolynomialRingOverFieldIsEuclideanDomain) of and is a constant polynomial, which can be normed to 1. Thus there exist the polynomials and of the ring such that
Hence, is a polynomial in with rational coefficients.
It follows that
whence (1) is true.
Suppose that we had also
with every rational. This implies that
with rational coefficients and degree less than . Because the degree of is , it is possible only if all differences vanish. Thus
i.e. the (1) is unique.
Note 1. The polynomial is called the canonical polynomial of the algebraic number with respect to the primitive element (http://planetmath.org/SimpleFieldExtension) .
Note 2. The theorem allows to denote the field similarly as polynomial rings:
Note 3. When allowed, unlike in (1), higher powers of the primitive element (whose minimal polynomial is ), one may unlimitedly write different sum of , e.g.
|Title||canonical form of element of number field|
|Date of creation||2013-03-22 19:08:00|
|Last modified on||2013-03-22 19:08:00|
|Last modified by||pahio (2872)|
|Defines||canonical form in number field|