algebraic sum and product

Let α,β be two elements of an extension fieldMathworldPlanetmath of a given field K.  Both these elements are algebraic over K if and only if both α+β and αβ are algebraic over K.

Proof.  Assume first that α and β are algebraicMathworldPlanetmath.  Because


and both here are finite (, then [K(α,β):K] is finite.  So we have a finite field extension K(α,β)/K which thus is also algebraic, and therefore the elements α+β and αβ of K(α,β) are algebraic over K.  Secondly suppose that α+β and αβ are algebraic over K.  The elements α and β are the roots of the quadratic equationx2-(α+β)x+αβ=0  (cf. properties of quadratic equation) with the coefficients in K(α+β,αβ).  Thus


Since  [K(α+β,αβ):K]  is finite,  then also  [K(α,β):K] is, and in the finite extension (  K(α,β)/K  the elements α and β must be algebraic over K.

Title algebraic sum and product
Canonical name AlgebraicSumAndProduct
Date of creation 2013-03-22 15:28:03
Last modified on 2013-03-22 15:28:03
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 11R32
Classification msc 11R04
Classification msc 13B05
Synonym sum and product algebraic
Related topic FiniteExtension
Related topic TheoryOfAlgebraicNumbers
Related topic FieldOfAlgebraicNumbers