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 -extension
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(Definition)
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Example 2 Let $p>2$ be a prime and, for any $n$ , let $\zeta_{p^n}$ be a primitive $p^n$ th root of unity. The cyclotomic extension: $$\Rats(\zeta_{p^n})/\Rats(\zeta_p)$$ is a $p$ -extension. Indeed: $$G_n=\operatorname{Gal}(\Rats(\zeta_{p^n})/\Rats)\cong (\Ints/p^n\Ints)^\times$$ Thus, $|G_n|=\varphi(p^n)=p^{(n-1)}(p-1)$ and $|G_1|=\varphi(p)=p-1$ , where $\varphi$ is the Euler phi function. Therefore the extension above is of degree $p^{(n-1)}$ .
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" -extension" is owned by alozano.
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Cross-references: degree, extension, Euler phi function, cyclotomic extension, root of unity, primitive, prime, field extension, integer, square-free, fields, Galois extension, prime number
There are 2 references to this entry.
This is version 2 of  -extension, born on 2005-02-17, modified 2005-02-17.
Object id is 6764, canonical name is PExtension.
Accessed 2558 times total.
Classification:
| AMS MSC: | 12F05 (Field theory and polynomials :: Field extensions :: Algebraic extensions) |
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Pending Errata and Addenda
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