Intereting Posts

Zero-dimensional ideals and finite-dimensional algebras
Understanding a proof of Komlós's theorem
Determine if a point is inside a subtriangle by its barycentric coordinates
Infinite product of measurable spaces
sum of irrational numbers – are there nontrivial examples?
Why: A holomorphic function with constant magnitude must be constant.
Are polynomials dense in Gaussian Sobolev space?
Is there a name for the “famous” inequality $1+x \leq e^x$?
canonical map of a monoid to its classifying space
On $x^3+y^3=z^3$, the Dixonian elliptic functions, and the Borwein cubic theta functions
Polynomials such that roots=coefficients
Understanding differentials
Asking about $M(q^2)$ and its order
$S_n = x_1^3+x_2^3+ \cdots +x_n^3$ squares perfect
A square of a rational between two positive real numbers ?!

Is the following proposition true? If yes, how would you prove this?

**Proposition**

Let $k$ be an algebraic number field.

Let $K$ be a finite abelian extension of $k$.

Suppose every principal prime ideal of $k$ splits completely in $K$.

Let $L$ be a finite extension of $k$.

Let $E = KL$.

Let $h’$ be the class number of $L$.

Then [$E : L$] | $h’$ and $E/L$ is unramified at every prime ideal of $L$.

- Can a prime in a Dedekind domain be contained in the union of the other prime ideals?
- Which number fields can appear as subfields of a finite-dimensional division algebra over Q with center Q?
- Proving the Kronecker Weber Theorem for Quadratic Extensions
- How to determine a Hilbert class field?
- Decomposition of a primitive regular ideal of a quadratic order
- Computing a uniformizer in a totally ramified extension of $\mathbb{Q}_p$.

**Motivation**

I thought I could use this proposition to prove the following result.

On the class number of a cyclotomic number field of an odd prime order

**Effort**

Let $\mathcal{I}$ be the group of fractional ideals of $L$.

Let $\mathcal{P}$ be the group of principal ideals of $L$.

Let $\mathcal{H}$ = {$I \in \mathcal{I}$; $N_{L/k}(I)$ is principal}.

Note that $\mathcal{H} \supset \mathcal{P}$.

Then use the following two links.

**Related questions**

On a certain criterion for unramification of an abelian extension of an algebraic number field

Complete splitting of a prime ideal in a certain abelian extension of an algebraic number field

- Basis for rank $n$ ring containing $1$.
- Show that the $\mathbb{Z}$-span $\mathbb{Z}b'_1+\cdots+ \mathbb{Z}b'_d$ of $B^+$ does not depend on the choice of $B$
- wildly ramified extensions and $p$-power roots of unity
- Exercise 1.10 from Silverman “The Arithmetic of Elliptic Curves ”
- Volume of first cohomology of arithmetic complex
- Where can the original paper by Takagi in English be found?
- What is so special about negative numbers $m$, $\mathbb{Z}$?
- Computing the ring of integers of a number field $K/\mathbb{Q}$: Is $\mathcal{O}_K$ always equal to $\mathbb{Z}$ for some $\alpha$?
- Ideals in a Quadratic Number Fields
- Is there an efficient algorithm to compute a minimal polynomial for the root of a polynomial with algebraic coefficients?

This is a direct application of class field theory. The assumption on $K$ implies that $K$ is contained in the Hilbert class field of $k$. Thus $E$ is contained in the Hilbert class field of $L$, and so $E$ is unramified over $L$, and $[E:L]$ divides the class number of $L$.

- Can it be determined that the sum of the diagonal entries, of matrix A, equals the sum of eigenvalues of A
- Open subsets in a manifold as submanifold of the same dimension?
- Finding all positive integers which satisfy $x^2-10y^2=1$
- Cube of harmonic mean
- Two circles overlap?
- Does X have a lower bound?
- Numerical Approximation of Differential Equations with Midpoint Method
- Non-Scientific questions solved by mathematics
- What's the difference between early transcendentals and late transcendentals?
- CDF of probability distribution with replacement
- Is $L^p \cap L^q$ dense in $L^r$?
- Is there a general formula for three Pythagorean Triangles which share an area?
- Subsets of finite sets of linearly independent vectors are linearly independent
- How to develop intuition in topology?
- Number of distinct numbers picked after $k$ rounds of picking numbers with repetition from $$