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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
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@awllower Thanks for your thoughtfulness. I am checking them out. –  Andrew Feb 10 '13 at 15:25
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1 Answer

up vote 5 down vote accepted

These are just group theoretic considerations and don't have anything to do specifically with ideal class groups.

  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 $ka + h_F b = 1$.

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Thanks very much. –  Andrew Feb 10 '13 at 0:44
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