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## Homework Statement

I need to prove that [itex]Q[√2,√3] = \{a+b√2+c√3+d√6: a, b, c, d \in Q\} [/itex] is a field.

## Homework Equations

## The Attempt at a Solution

I proved all axioms except the existence of inverses for nonzero elements. My problem is that multiplication is quite hairy. I ended up trying to invert a 4x4 matrix with letters, which got tedious very quickly and I couldn't finish it. Actually, proving that the determinant is never 0 would be good enough since this would prove linear independence and the existence of a unique solution.

Is there a much simpler method, or do I really have to deal with ugly algebra?