splitting and ramification in number fields and Galois extensions

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 (http://planetmath.org/FractionalIdeal)). 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}{{\mathfrak{P}}_{i}}% ^{e_{i}}

(1)

We say that the primes ${\mathfrak{P}}_{i}$ lie above ${\mathfrak{p}}$ and ${\mathfrak{P}}_{i}|{\mathfrak{p}}$ (divides). The exponent $e_{i}$ (commonly denoted as $e({\mathfrak{P}}_{i}|{\mathfrak{p}})$ ) is the ramification index of ${\mathfrak{P}}_{i}$ over ${\mathfrak{p}}$ . Notice that for each prime ideal ${\mathfrak{P}}_{i}$ , the quotient ring $\mathcal{O}_{F}/{\mathfrak{P}}_{i}$ 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 ${\mathfrak{P}}_{i}$ over ${\mathfrak{p}}$ and it is usually denoted by:

f({\mathfrak{P}}_{i}|{\mathfrak{p}})=[\mathcal{O}_{F}/{\mathfrak{P}}_{i}:% \mathcal{O}_{K}/{\mathfrak{p}}].

Notice that as it is pointed out in the entry “inertial degree (http://planetmath.org/InertialDegree)”, the ramification index and the inertial degree are related by the formula:

\displaystyle\sum_{i=1}^{r}e({\mathfrak{P}}_{i}|{\mathfrak{p}})f({\mathfrak{P}% }_{i}|{\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 ${\mathfrak{P}}_{i},{\mathfrak{p}}$ be as above.

1.

If $e_{i}>1$ for some $i$ , then we say that ${\mathfrak{P}}_{i}$ 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 ${\mathfrak{P}}$ lying above ${\mathfrak{p}}$ (so $r=1$ ) and $f({\mathfrak{P}}|{\mathfrak{p}})=1$ then we say that ${\mathfrak{p}}$ is totally ramified in $F/K$ . In this case $e({\mathfrak{P}}|{\mathfrak{p}})=[F:K]$ .
3.

On the other hand, if $e({\mathfrak{P}}_{i}|{\mathfrak{p}})=f({\mathfrak{P}}_{i}|{\mathfrak{p}})=1$ 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 $e_{i}=e({\mathfrak{P}}_{i}|{\mathfrak{p}})>1$ and $e_{i}$ and $p$ are relatively prime, then we say that ${\mathfrak{P}}_{i}$ is tamely ramified. If $p|e_{i}$ then we say that ${\mathfrak{P}}_{i}$ 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 $e_{i}=e({\mathfrak{P}}_{i}|{\mathfrak{p}})$ are equal to the same number $e$ , all the inertial degrees $f_{i}=f({\mathfrak{P}}_{i}|{\mathfrak{p}})$ are equal to the same number $f$ and the ideal ${\mathfrak{p}}\mathcal{O}_{F}$ factors as:

{\mathfrak{p}}\mathcal{O}_{F}=\prod_{i=1}^{r}{\mathfrak{P}}_{i}^{e}=({% \mathfrak{P}}_{1}\cdot{\mathfrak{P}}_{2}\cdot\ldots\cdot{\mathfrak{P}}_{r})^{e}

Moreover:

e\cdot f\cdot r=[F:K].

Title	splitting and ramification in number fields and Galois extensions
Canonical name	SplittingAndRamificationInNumberFieldsAndGaloisExtensions
Date of creation	2013-03-22 15:05:29
Last modified on	2013-03-22 15:05:29
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	11
Author	alozano (2414)
Entry type	Definition
Classification	msc 12F99
Classification	msc 13B02
Classification	msc 11S15
Synonym	completely split
Synonym	strongly ramified
Synonym	wild ramification
Related topic	Ramify
Related topic	InertialDegree
Related topic	CalculatingTheSplittingOfPrimes
Related topic	PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ
Related topic	PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ
Defines	totally ramified
Defines	totally split
Defines	wildly ramified
Defines	tamely ramified