Abstract Algebra: Theory and Applications · Problem 20.3

Fields — Thomas Judson

Mathematics — Abstract Algebra · vector-spaces-basis-dimension

Problem

Let

\mathbb{Q}(\sqrt{2}, \sqrt{3})

be the field generated by elements of the form

a + b\sqrt{2} + c\sqrt{3} + d\sqrt{6}

, where

a, b, c, d

are in

\mathbb{Q}

. Prove that

\mathbb{Q}(\sqrt{2}, \sqrt{3})

is a vector space of dimension 4 over

\mathbb{Q}

. Find a basis for

\mathbb{Q}(\sqrt{2}, \sqrt{3})

. Hint.

\mathbb{Q}(\sqrt{2}, \sqrt{3})

has basis

\{1, \sqrt{2}, \sqrt{3}, \sqrt{6}\}

over

\mathbb{Q}