Need help understanding splitting fields

[PsychonautQQ]

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 want to shed some light on my situation?

 

[WWGD]

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]

Ah, sorry, I misread, let me think again.

 

[HallsofIvy]

It looks to me like a misprint. No, the extension fields containing p^{1/2} and q^{1/2}, for p and q prime, are NOT the same.