|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
an exact sequence for ray class groups
|
(Result)
|
|
|
Let $K$ be a number field, let $\mathcal{O}_K$ be its ring of integers and let $\mathfrak{m}$ be a modulus in $K$ , i.e. $$\mathfrak{m}=\mathfrak{m}_0\mathfrak{m}_\infty$$ where $\mathfrak{m}_0$ is an integral ideal in $\mathcal{O}_K$ and $\mathfrak{m}_\infty$ is a product of real infinite places (i.e. real archimedean primes). Let $\Cl(K)$ be the ideal class group of $K$ and let $\Cl(K,\mathfrak{m})$ be the ray class group of $K$ of conductor
$\mathfrak{m}$ . Also, define $$(\mathcal{O}_K/\mathfrak{m})^\times=(\mathcal{O}_K/\mathfrak{m}_0)^\times \times (\Ints/2\Ints)^{|\mathfrak{m}_\infty|}$$ where $|\mathfrak{m}_\infty|$ denotes the number of real places in $\mathfrak{m}$ . Finally, let $U=\mathcal{O}_K^\times$ be the unit group of $K$ .
Proposition 1 The elements above fit in the following exact sequence: $$U\longrightarrow (\mathcal{O}_K/\mathfrak{m})^\times\longrightarrow \Cl(K,\mathfrak{m})\longrightarrow \Cl(K)\longrightarrow 1.$$
Example 1 Let $K=\Rats$ . Thus, $\Cl(\Rats)$ is trivial and $U=\{ \pm 1\}\cong \Ints/2\Ints$ . Let $\mathfrak{m}=p\infty$ where $p>2$ is any prime. Then: $$(\Ints/\mathfrak{m})^\times=(\Ints/p\Ints)^\times \times (\Ints/2\Ints).$$ The exact sequence now reads: $$\Ints/2\Ints\longrightarrow (\Ints/p\Ints)^\times \times (\Ints/2\Ints)\longrightarrow \Cl(\Rats,p\infty)\longrightarrow 1.$$ Therefore, $\Cl(\Rats,p\infty)\cong (\Ints/p\Ints)^\times$ . In fact, as we know, the ray class field of $\Rats$ of conductor $\mathfrak{m}=p\infty$ is the cyclotomic field $\Rats(\zeta_p)$ where $\zeta_p$ is any primitive $p$ th root of unity. Moreover $$\Gal(\Rats(\zeta_p)/\Rats)\cong \Cl(\Rats,p\infty)\cong (\Ints/p\Ints)^\times.$$ Finally notice that the ray class group of $\Rats$ of conductor $\mathfrak{m}=p$ is simply $(\Ints/p\Ints)^\times/\{\pm 1 \}$ which corresponds to the ray class field $\Rats(\zeta_p)^+=\Rats(\zeta_p+\zeta_p^{-1})$ , the maximal real
subfield of $\Rats(\zeta_p)$ .
|
"an exact sequence for ray class groups" is owned by alozano.
|
|
(view preamble | get metadata)
Cross-references: maximal real subfield, root of unity, primitive, cyclotomic field, exact sequence, unit group, places, number, conductor, ray class group, ideal class group, primes, archimedean, infinite places, real, product, integral ideal, modulus, ring of integers, number field
This is version 3 of an exact sequence for ray class groups, born on 2006-02-28, modified 2006-02-28.
Object id is 7660, canonical name is AnExactSequenceForRayClassGroups.
Accessed 1133 times total.
Classification:
| AMS MSC: | 11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
