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$p$-extension (Definition)
Definition 1   Let $ p$ be a prime number. A Galois extension of fields $ E/F$, with $ G=\operatorname{Gal}(E/F)$, is said to be a $ p$-extension if $ G$ is a $ p$-group.
Example 1   Let $ d$ be a square-free integer. Then the field extension $ \mathbb{Q}(\sqrt{d})/\mathbb{Q}$ is a $ 2$-extension.
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:
$\displaystyle \mathbb{Q}(\zeta_{p^n})/\mathbb{Q}(\zeta_p)$
is a $ p$-extension. Indeed:
$\displaystyle G_n=\operatorname{Gal}(\mathbb{Q}(\zeta_{p^n})/\mathbb{Q})\cong (\mathbb{Z}/p^n\mathbb{Z})^\times$
Thus, $ \vert G_n\vert=\varphi(p^n)=p^{(n-1)}(p-1)$ and $ \vert G_1\vert=\varphi(p)=p-1$, where $ \varphi$ is the Euler phi function. Therefore the extension above is of degree $ p^{(n-1)}$.



"$p$-extension" is owned by alozano.
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See Also: $p$-group, unramified extensions and class number divisibility, push-down theorem on class numbers, class number divisibility in $p$-extensions, quadratic extension

Other names:  p-extension
Keywords:  field extension
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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
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This is version 2 of $p$-extension, born on 2005-02-17, modified 2005-02-17.
Object id is 6764, canonical name is PExtension.
Accessed 2093 times total.

Classification:
AMS MSC: 12F05 (Field theory and polynomials :: Field extensions :: Algebraic extensions)
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