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For example

Q(sqrt(2)+sqrt(3))

I know that if i take a=sqrt(2)+sqrt(3) that i can find a polynomial (1/4)x^4 - (5/2)x^2 + 1/4 that when evaluated at a is equal to zero.

So, from that I know the degree is 4 and the basis should have 4 dimensions.

{1, sqrt(2), sqrt(3), Y} where Y is the part I don't understand.

{1, sqrt(2), sqrt(3), sqrt(6)} is the actual basis, but how do you get sqrt(6) as Y?

I don't see how it is a linear combination of sqrt(2) and sqrt(3) as defined in the book I'm using. Since it should be a additive combination of scalars from Q times sqrt(2) and sqrt(3)... but the only way to obtain sqrt(6) is sqrt(2)*sqrt(3), in which neither are scalars as elements of Q.

Any help understanding this concept is greatly appreciated.