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June 18th, 2008, 06:37 AM   #1
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Degree of an extension

Given the polynomial f(t)=t^4 - 2 over Q, and L the splitting field of f, how does one determine the degree of the extension L:Q?
If anyone could help me out with this i would be very appreciated.Thank you.
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June 24th, 2008, 09:58 PM   #2
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Re: Degree of an extension

Obviously s:= 2^(1/4) is a root, and the other ones are s*w, with w a 4-th root of unity in C ==> if we put w = i, then

the roots are s, s*w= s*i, s*w^2 = -s, and s*w^3 = -s*i.

Now Q(s,i) is clearly the splitting field of the pol. over Q, and you can write Q <= Q(s) <= Q(s,i), and thus you can now calculate the extension's degree.

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Tonio
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