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How is this possible? (in other words, why are the powers different) K= Q( i, sqrt(2))...

How is this possible? (in other words, why are the powers different)

K= Q( i, sqrt(2)) is the root field over Q of x^4 - 2x^2 + 9, and it's the root field over Q(sqrt(2) of x^2 - 2(sqrt(2))x + 3

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Answer #1

Different powers can have same root field because root field depends upon roots of polynomial not on degree of polynomial...two polynomials with different degrees can have same kind of roots although with different multiplicities...

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