algebraic sum and product
Let be two elements of an extension field of a given field . Both these elements are algebraic over if and only if both and are algebraic over .
Proof. Assume first that and are algebraic. Because
and both here are finite (http://planetmath.org/ExtendedRealNumbers), then is finite. So we have a finite field extension which thus is also algebraic, and therefore the elements and of are algebraic over . Secondly suppose that and are algebraic over . The elements and are the roots of the quadratic equation (cf. properties of quadratic equation) with the coefficients in . Thus
Since is finite, then also is, and in the finite extension (http://planetmath.org/FiniteExtension) the elements and must be algebraic over .
|Title||algebraic sum and product|
|Date of creation||2013-03-22 15:28:03|
|Last modified on||2013-03-22 15:28:03|
|Last modified by||pahio (2872)|
|Synonym||sum and product algebraic|