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[parent] Hermite's theorem (Corollary)

The following is a corollary of Minkowski's theorem on ideal classes, which is a corollary of Minkowski's theorem on lattices.

Definition 1   Let $ S=\{p_1,\ldots,p_r\}$ be a set of rational primes $ p_i \in \mathbb{Z}$. We say that a number field $ K$ is unramified outside $ S$ if any prime not in $ S$ is unramified in $ K$. In other words, if $ p$ is ramified in $ K$, then $ p\in S$. In other words, the only primes that divide the discriminant of $ K$ are elements of $ S$.
Corollary 1 (Hermite's Theorem)   Let $ S=\{p_1,\ldots,p_r\}$ be a set of rational primes $ p_i \in \mathbb{Z}$ and let $ N\in \mathbb{N}$ be arbitrary. There is only a finite number of fields $ K$ which are unramified outside $ S$ and bounded degree $ [K:\mathbb{Q}]\leq N$.



"Hermite's theorem" is owned by alozano.
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Also defines:  unramified outside a set of primes

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Cross-references: degree, bounded, fields, number, finite, discriminant, divide, prime, unramified, number field, rational primes, Minkowski's theorem, Minkowski's theorem on ideal classes
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This is version 2 of Hermite's theorem, born on 2005-02-24, modified 2006-09-26.
Object id is 6820, canonical name is HermitesTheorem.
Accessed 2435 times total.

Classification:
AMS MSC: 11H06 (Number theory :: Geometry of numbers :: Lattices and convex bodies)
 11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants)
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