# Splitting fields

• Mar 24th 2010, 06:12 AM
bobisback
Splitting fields
determine the splitting field K/Q and calculate [K:Q]

i) $X^{4}-5$
I think i understand just need reasurance

spliting field $(x^{2}-\sqrt{5})(x^{2}+\sqrt{5})$

[K:Q] =4 because $x^{4} - 5$ is min polynomial

2) $x^{4}-7x^{2}+10$

splitting field $(x^{2}-5)(x^{2}-2)$

[K:Q] = 4

is this all correct?

thanks bobisback
• Mar 24th 2010, 09:11 AM
tonio
Quote:

Originally Posted by bobisback
determine the splitting field K/Q and calculate [K:Q]

i) $X^{4}-5$
I think i understand just need reasurance

spliting field $(x^{2}-\sqrt{5})(x^{2}+\sqrt{5})$

[K:Q] =4 because $x^{4} - 5$ is min polynomial

But you haven't yet determined $K$ , and its degree is wrong: certainly $[\mathbb{Q}(\sqrt[4]{5}):\mathbb{Q}]=4$ , but this extension field of the rationals is NOT

the splitting field of the given pol. over the rationals since it doesn't contain ALL its roots...for example, $\sqrt[4]{5}\,i$ is a root of the pol. that isn't contained in $\mathbb{Q}(\sqrt[4]{5})$.

What you did is just fine to find the roots: $x^4-5=(x^2-\sqrt{5})(x^2+\sqrt{5})$ , but then it must be clear that both roots of the rightmost factor are complex

non-real ( in fact, a conjugate pair: $\pm \sqrt[4]{5}\,i$ ) , so they aren't contained in a real field as $\mathbb{Q}(\sqrt[4]{5})$ .

Take it from here and try now to find explicitly what $K$ is and its degree over the rationals.

2) $x^{4}-7x^{2}+10$

splitting field $(x^{2}-5)(x^{2}-2)$

[K:Q] = 4

is this all correct?

This time the degree is correct (though you gave no explanation at all...(Doh) ) but again you haven't explicitly described the splitting field $K$ ...

Tonio

thanks bobisback

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