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[parent] splitting and ramification in number fields and Galois extensions (Definition)

Let $ F/K$ be an extension of number fields and let $ \mathcal{O}_F$ and $ \mathcal{O}_K$ be their respective rings of integers. The ring of integers of a number field is a Dedekind domain, and these enjoy the property that every ideal $ {\mathfrak{A}}$ factors uniquely as a finite product of prime ideals (see the entry fractional ideal). Let $ {\mathfrak{p}}$ be a prime ideal of $ \mathcal{O}_K$. Then $ {\mathfrak{p}}\mathcal{O}_F$ is an ideal of $ \mathcal{O}_F$. Let us assume that the prime ideal factorization of $ {\mathfrak{p}}\mathcal{O}_F$ into primes of $ \mathcal{O}_F$ is as follows:

$\displaystyle {\mathfrak{p}}\mathcal{O}_F=\prod_{i=1}^r {\P _i}^{e_i}$     (1)

We say that the primes $ \P _i$ lie above $ {\mathfrak{p}}$ and $ \P _i\vert{\mathfrak{p}}$ (divides). The exponent $ e_i$ (commonly denoted as $ e(\P _i\vert{\mathfrak{p}})$) is the ramification index of $ \P _i$ over $ {\mathfrak{p}}$. Notice that for each prime ideal $ \P _i$, the quotient ring $ \mathcal{O}_F/\P _i$ is a finite field extension of the finite field $ \mathcal{O}_K/{\mathfrak{p}}$ (also called the residue field). The degree of this extension is called the inertial degree of $ \P _i$ over $ {\mathfrak{p}}$ and it is usually denoted by:
$\displaystyle f(\P _i\vert{\mathfrak{p}})=[\mathcal{O}_F/\P _i:\mathcal{O}_K/{\mathfrak{p}}].$

Notice that as it is pointed out in the entry “inertial degree”, the ramification index and the inertial degree are related by the formula:

$\displaystyle \sum_{i=1}^r e(\P _i\vert{\mathfrak{p}})f(\P _i\vert{\mathfrak{p}})=[F:K]$     (2)

where $ r$ is the number of prime ideals lying above $ {\mathfrak{p}}$ (as in Eq. (1)). See the theorem below for an improvement of Eq. (2) in the case when $ F/K$ is Galois.
Definition 1   Let $ F,K$ and $ \P _i,{\mathfrak{p}}$ be as above.
  1. If $ e_i>1$ for some $ i$, then we say that $ \P _i$ is ramified over $ {\mathfrak{p}}$ and $ {\mathfrak{p}}$ ramifies in $ F/K$. If $ e_i=1$ for all $ i$ then we say that $ {\mathfrak{p}}$ is unramified in $ F/K$.
  2. If there is a unique prime ideal $ \P$ lying above $ {\mathfrak{p}}$ (so $ r=1$) and $ f(\P \vert{\mathfrak{p}})=1$ then we say that $ {\mathfrak{p}}$ is totally ramified in $ F/K$. In this case $ e(\P \vert{\mathfrak{p}})=[F:K]$.
  3. On the other hand, if $ e(\P _i\vert{\mathfrak{p}})=f(\P _i\vert{\mathfrak{p}})=1$ for all $ i$, we say that $ {\mathfrak{p}}$ is totally split (or splits completely) in $ F/K$. Notice that there are exactly $ r=[F:K]$ prime ideals of $ \mathcal{O}_F$ lying above $ {\mathfrak{p}}$.
  4. Let $ p$ be the characteristic of the residue field $ \mathcal{O}_K/{\mathfrak{p}}$. If $ e_i=e(\P _i\vert{\mathfrak{p}})>1$ and $ e_i$ and $ p$ are relatively prime, then we say that $ \P _i$ is tamely ramified. If $ p\vert e_i$ then we say that $ \P _i$ is strongly ramified (or wildly ramified).

When the extension $ F/K$ is a Galois extension then Eq. (2) is quite more simple:

Theorem 1   Assume that $ F/K$ is a Galois extension of number fields. Then all the ramification indices $ e_i=e(\P _i\vert{\mathfrak{p}})$ are equal to the same number $ e$, all the inertial degrees $ f_i=f(\P _i\vert{\mathfrak{p}})$ are equal to the same number $ f$ and the ideal $ {\mathfrak{p}}\mathcal{O}_F$ factors as:
$\displaystyle {\mathfrak{p}}\mathcal{O}_F= \prod_{i=1}^r \P _i^e=(\P _1\cdot\P _2\cdot\ldots\cdot\P _r)^e$
Moreover:
$\displaystyle e\cdot f\cdot r=[F:K].$



"splitting and ramification in number fields and Galois extensions" is owned by alozano.
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See Also: ramification index, inertial degree, calculating the splitting of primes, prime ideal decomposition in quadratic extensions of $\mathbb{Q}$, prime ideal decomposition in cyclotomic extensions of $\mathbb{Q}$

Other names:  completely split, strongly ramified, wild ramification
Also defines:  totally ramified, totally split, wildly ramified, tamely ramified

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the ramification index and the inertial degree are multiplicative in towers (Theorem) by alozano
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Cross-references: indices, simple, Galois extension, relatively prime, characteristic, inertial degree, degree, residue field, finite field, finite field extension, quotient ring, ramification index, exponent, primes, prime ideals, product, factors, ideal, property, Dedekind domain, rings of integers, number fields, extension
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This is version 8 of splitting and ramification in number fields and Galois extensions, born on 2005-02-23, modified 2007-05-10.
Object id is 6818, canonical name is SplittingAndRamificationInNumberFieldsAndGaloisExtensions.
Accessed 6797 times total.

Classification:
AMS MSC: 11S15 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Ramification and extension theory)
 13B02 (Commutative rings and algebras :: Ring extensions and related topics :: Extension theory)
 12F99 (Field theory and polynomials :: Field extensions :: Miscellaneous)
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