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− | = | + | = Problem 5 = |
+ | Clearly <math>\mathbb{Q}(\alpha)\subset \mathbb{Q}(2^{\frac{1}{3}})</math>. But also, | ||
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+ | <math>2\alpha - \alpha = 16-17 \cdot 2^{\frac{1}{3}}</math> | ||
+ | So <math>\mathbb{Q}(2^{\frac{1}{3}}) \subset \mathbb{Q}(\alpha)</math> and <math>\mathbb{Q}(2^{\frac{1}{3}}) = \mathbb{Q}(\alpha)</math>. | ||
− | + | Thus <math>3=|\mathbb{Q}(2^{\frac{1}{3}}):\mathbb{Q}| = |\mathbb{Q}(\alpha):\mathbb{Q}|</math>. | |
+ | <br> | ||
+ | [[NinjaSharksSet5|Back to NinjaSharksSet5]] | ||
− | + | [[Category:NinjaSharksSet5]] | |
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Latest revision as of 04:11, 3 July 2013
Problem 5
Clearly $ \mathbb{Q}(\alpha)\subset \mathbb{Q}(2^{\frac{1}{3}}) $. But also,
$ 2\alpha - \alpha = 16-17 \cdot 2^{\frac{1}{3}} $
So $ \mathbb{Q}(2^{\frac{1}{3}}) \subset \mathbb{Q}(\alpha) $ and $ \mathbb{Q}(2^{\frac{1}{3}}) = \mathbb{Q}(\alpha) $.
Thus $ 3=|\mathbb{Q}(2^{\frac{1}{3}}):\mathbb{Q}| = |\mathbb{Q}(\alpha):\mathbb{Q}| $.