- #1

PsychonautQQ

- 784

- 10

## Homework Statement

let b be a square root of 1+i, show that Q(b):Q is not a normal extension. Also, what is the Galois group of the extension?

## Homework Equations

## The Attempt at a Solution

so b = +/- (1+i)^(1/2), and it's minimal polynomial is x^4+4 which has roots -(2)^1/2 and 2^(1/2) that are not in Q(b) and therefore the extension is not normal. In a proper splitting field, x^4+4 splits into (x-(1+i))(x+(1+i))(x-2^(1/2))(x+2^(1/2)), the Galois group would have a map that permutes the roots of the first two factors and a map that would permute the roots of the second two factors, therefore it would be isomorphic to Z_2 x Z_2.

Is this correct?