# Beginner questions regarding $Cl_F$, $h_F$, and principal ideals in $D_F$

I would appreciate help (as a self-studier) with what are some very beginner questions regarding the class group $Cl_F$ and its class number $h_F$. They are related to Ch. 12 in "Ireland & Rosen" and a set of online class notes that are close to it.

"I&R" defines equivalence as: for non zero ideals $I, J \subseteq D_F$ ($D_F$ is the ring of integers) there exist $a,b \in D_F$ such that $a I = b J$. If possible, I would appreciate any of your kind remarks to use this definition.

-- If $h_F$ is the class number, what is the order of the class group? I ask this because in the proof of Prop 12.2.5 which is:

For any ideal $A \subset D$ ($D$ is the ring of integers) there is an integer $k$,

$1\le k \le h_F$ such that $A^{k}$ is principal.

My question comes in the beginning of the proof:

"Consider the set of ideals {$A^{i}$: $1\le i \le h_F + 1$}" Why does the power of $A$ go to $h_F + 1$ rather than just $h_F$?

-- My second question is: F is a number field, $D_F$ its ring of integers, $I, J \subseteq D_F$ non-zero ideals. Let $[I]$ be its equivalence class, with a product operation defined on $Cl_F$ by $[I] [J] = [IJ]$.This makes $Cl_F$ into a finite, abelian group with identity element $D_F$.

Show $I^{h_F}$ is a principal ideal. I would guess one would want to show $I^{h_F}$ is in $[D_F]$ which is the equivalence class of principal ideals, but I would appreciate help since I am confused about the order of $Cl_F$ and also how to do it.

-- My last question is along similar (what I think are group theoretic) lines.

If $I^{k}$ is principal for some $k$ prime to $h_F$, then $I$ itself is principal. So I guess I need to show that $I$ is in $[D_F]$. I remember that the residue of $k$ is a unit in $\mathbb{Z}/h_F \mathbb{Z}$ and generates it as an additive group. But even if this is correct, I would appreciate help.

P.S. If there are any references at this elementary level, I would appreciate any recommendations.

Thanks.

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A suggestion. The book does a particularly intriguing and inspiring work on the intruduction to cyclotomic fields and its properties. Of course this contains more information on this subject. – awllower Feb 10 '13 at 13:35
@awllower Thanks for your thoughtfulness. I am checking them out. – TheBirdistheWord Feb 10 '13 at 15:25

1. What is the order of the class group? It is just the class number, $h_F$. The class number is defined as the order of the class group.
2. Why does the power of $A$ go to $h_F+1$ rather than just $h_F$? Well, it depends on what the next line in the proof is, but I'm guessing it's something like "Since there are only $h_F$ equivalence classes of ideals, but $h_F+1$ integers from $1,2,\ldots,h_F+1$, there must exist 2 distinct integers $i$ and $j$ between 1 and $h_F + 1$ such that $[A^i] = [A^j]$." You need $h_F+1$ values so that you have more possible exponents than you have equivalence classes of ideals (pigeonhole principle: If $n+1$ pigeons are placed in $n$ boxes, then at least one box contains 2 pigeons).
3. Show that $I^{h_F}$ is a principal ideal. This follows from Lagrange's Theorem applied to the cyclic subgroup $\{1, I, I^2, ... , I^{k-1}\}$ generated by $I$. Here $k$ is the smallest positive integer such that $I^k$ is principal (i.e. equals 1, the equivalence class of principal ideals). By Lagrange's Theorem, $k$ is a divisor of $h_F$.
4. If $I^k$ is principal and $k$ is relatively prime to $h_F$, then $I$ is principal. Use the fact that (because $k$ and $h_F$ are relatively prime), there exist integers $a$ and $b$ such that $ka + h_F b = 1$.