# Need help understanding splitting fields

My textbook is going through an example on splitting fields. It asked to find a splitting field for x^4 - 6x^2 - 7 over the rational numbers. This polynomial factors to (x^2 - 7)*(x^2+1) which has roots of 7^(1/2) and i. So i figured the extension field E we are looking for is Q(i)(7^(1/2)), but my textbook jumps straight to Q(i)(2^(1/2).

is the squareroot of 7 an element of the simple extension of the rational numbers with the square root of two? I can't imagine it being (yet i can't imagine many things that are..).

Any mathamaverick wanna shed some light on my situation?

WWGD
Gold Member
Doesn't it follow, if ##2^{1/2}## is in a field extension of the rationals that ##(7/2)( 2^{1/2}) ## is also on the field, e.g., by closure under multiplication? Your result is correct if you re not looking for a minimal extension.

• PsychonautQQ
Okay, what am i missing? How does (7/2)*(2^(1/2)) show that the finite extensions of 7^(1/2) reduces to 2^(1/2)?

WWGD
Gold Member
Ah, sorry, I misread, let me think again.

• PsychonautQQ
HallsofIvy
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