Let be a field of characteristic not 2, and let and be non-squares in . Prove that has degree 4 over if is nonsquare in , and degree 2 otherwise. (If the degree is 4, is called a biquadratic extension of .
Suppose is nonsquare in . We claim that is irreducible over . To see this, suppose to the contrary that . Comparing coefficients, we have and . If , then , a contradiction. If , then . But then is square, a contradiction. So has degree 2 over , and thus degree 4 over .
Now suppose that is square in . then is reducible over , and we have of degree 2 over .