In page 22 from Neukirch's Algebraic Number Theory, he defines the class group $Cl_K$ of a number field $K$ to be the quotient of group of fractional ideals $J_K$ by the subgroup of principal ideals $P_K$.

He shows the following exact sequence: $$1\to \mathcal{O}^*\to K^*\to J_K\to Cl_K\to 1$$

Then he makes the following comment: "[...] the class group measures the expansion that takes place when we pass from numbers to ideals, whereas the unit group measures the contraction in the same process".

Two questions: 1) What does he mean by "expansion" and "contraction"? 2) How is this exact sequence illuminating in any sense?


Every element of $K^{\times}$ generates a principal fractional ideal, and two elements of $K^{\times}$ generate the same principal fractional ideal if they differ by a unit of $\mathcal{O}_K$, i.e. by an element of $\mathcal{O}^{\times}_K$. In this sense the unit group $\mathcal{O}_K^{\times}$ measures the proportion of numbers to principal fractional ideals. Here numbers is understood to mean elements of $K^{\times}$.

This characterization also shows that the group $P_K$ of principal fractional ideals of $K$ is precisely the cokernel of the map $$1\ \longrightarrow\ \mathcal{O}_K^{\times}\ \longrightarrow\ K^{\times}.\tag{1}$$ That is to say, the group $P_K$ fits into the short exact sequence $$1\ \longrightarrow\ \mathcal{O}_K^{\times}\ \longrightarrow\ K^{\times}\ \longrightarrow\ P_K\ \longrightarrow\ 1.\tag{2}$$

Of course not all fractional ideals are principal in general. That is to say, going from numbers to ideals we also gain new ideals, apart from those generated by single numbers (i.e. principal ideals). The proportion of fractional ideals to principal fractional ideals is measured by the quotient $J_K/P_K$. This quotient is called the class group of $K$, denoted by $\operatorname{Cl}_K$.

This can be phrased briefly by saying that the class group $\operatorname{Cl}_K$ of $K$ is the cokernel of the map $$1\ \longrightarrow\ P_K\ \longrightarrow\ J_K.\tag{3}$$ Then by definition $\operatorname{Cl}_K$ fits into the short exact sequence $$1\ \longrightarrow\ P_K\ \longrightarrow\ J_K\ \longrightarrow\ \operatorname{Cl}_K\ \longrightarrow\ 1.\tag{4}$$ Putting the two short exact sequence $(2)$ and $(4)$ together shows that $\operatorname{Cl}_K$ fits into the exact sequence $$1\ \longrightarrow\ \mathcal{O}_K^{\times}\ \longrightarrow\ K^{\times}\ \longrightarrow\ J_K\ \longrightarrow\ \operatorname{Cl}_K\ \longrightarrow\ 1.$$ This shows that the group of units and the class group are the kernel and cokernel, respectively, of the map $K^{\times}\ \longrightarrow\ J_K$. The former measures the how many numbers that contract to the same ideal, the latter measures the proportion of ideals to ideals coming from numbers.

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