properties of (ϑ)-conjugates

Lemma.  Let α1,α2,,αs be algebraic numbersMathworldPlanetmath belonging to the number fieldMathworldPlanetmath (ϑ) of degree ( n and αi(j) theirϑ)-conjugatesPlanetmathPlanetmath.  If P(x1,x2,,xs) is a polynomialPlanetmathPlanetmath with rational coefficients and if

P(α1,α2,,αs)= 0,

then also

P(α1(j),α2(j),,αs(j))= 0

for each  j=1, 2,,n.  In the special case of two elements α and β of (ϑ) one may infer the formulae

(αβ)(j)=α(j)β(j),(α+β)(j)=α(j)+β(j). (1)

The lemma implies easily the following theorems.

Theorem 1.  All conjugate fields of (ϑ) are isomorphic.

Theorem 2.  The norm and the trace in the field (ϑ) satisfy

Title properties of (ϑ)-conjugates
Canonical name PropertiesOfmathbbQvarthetaconjugates
Date of creation 2013-03-22 19:09:14
Last modified on 2013-03-22 19:09:14
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Topic
Classification msc 11R04
Classification msc 11C08
Classification msc 12E05
Classification msc 12F05
Related topic ConjugateFields
Related topic IndependenceOfCharacteristicPolynomialOnPrimitiveElement