# characterizing CM-fields using Dirichlet’s unit theorem

If $K$ is a number field^{}, ${\mathcal{O}}_{K}$ is the ring of algebraic integers in $K$, and ${\mathcal{O}}_{K}^{\star}$ is the (multiplicative) group of units in ${\mathcal{O}}_{K}$. Dirichlet’s unit theorem gives the structure^{} of the unit group. We can use that theorem to characterize CM-fields:

###### Theorem 1.

Let $\mathrm{Q}\mathrm{\subset}F\mathrm{\subset}K$ be nontrivial extensions^{} of number fields. Then $K$ is a CM-field, with $F$ its totally real subfield^{}, if and only if ${\mathrm{O}}_{K}^{\mathrm{\star}}\mathrm{/}{\mathrm{O}}_{F}^{\mathrm{\star}}$ is finite.

We use the notation of the article on Dirichlet’s unit theorem, where $r$ (and ${r}_{F},{r}_{K}$) is used to count real embeddings and $s$ (as well as ${s}_{F},{s}_{K}$) to count complex embeddings, and we write $\mu (F)$ or $\mu (K)$ for the group of roots of unity^{} in ${\mathcal{O}}_{F}^{\star}$ or ${\mathcal{O}}_{K}^{\star}$.

Proof.

Write $n=[F:\mathbb{Q}],m=[K:F]>1$.

($\Rightarrow $): If $K/F$ is CM, then since $F$ is totally real, ${r}_{F}=n,{s}_{F}=0$. Hence by Dirichlet’s unit theorem, ${\mathcal{O}}_{F}^{\star}\cong \mu (F)\times {\mathbb{Z}}^{n-1}$. Since $K/F$ is a complex quadratic extension, $[K:\mathbb{Q}]=2n$ and all its embeddings^{} are complex. Thus ${r}_{K}=0,\mathrm{\hspace{0.25em}2}{s}_{K}=2n$. Hence ${\mathcal{O}}_{K}^{\star}\cong \mu (K)\times {\mathbb{Z}}^{n-1}$ as well. Clearly ${\mathcal{O}}_{F}^{\star}\subset {\mathcal{O}}_{K}^{\star}$, and since they have the same rank (http://planetmath.org/FreeModule), their quotient is torsion and thus finite.

($\Leftarrow $): Since ${\mathcal{O}}_{K}^{\star}/{\mathcal{O}}_{F}^{\star}$ is finite, the ranks of these groups are equal and thus ${r}_{F}+{s}_{F}={r}_{K}+{s}_{K}$ again by Dirichlet’s unit theorem.

Now,

${r}_{K}+2{s}_{K}$ | $=mn=m({r}_{F}+2{s}_{F})$ | (1) | ||

${r}_{K}+{s}_{K}$ | $={r}_{F}+{s}_{F};$ | (2) |

subtracting (2) from (1), we get

$${s}_{K}=(m-1)({r}_{F}+2{s}_{F})+{s}_{F}\ge (m-1)n$$ | (3) |

and thus $mn={r}_{K}+2{s}_{K}\ge {r}_{K}+2(m-1)n$ so that $0\le {r}_{K}\le n(2-m)$. Thus $m\le 2$, and since $K$ is a nontrivial extension, we must have $m=2$ so that $K/F$ is quadratic and ${r}_{K}=0$ (since $n(2-m)=0$).

Finally, by (3), we then have ${s}_{K}={r}_{F}+3{s}_{F}$; (2) says that ${s}_{K}={r}_{F}+{s}_{F}$, and thus ${s}_{F}=0$. It follows that $F$ is totally real and, since ${r}_{K}=0$, $K$ must be an imaginary quadratic extension of $F$.

Title | characterizing CM-fields using Dirichlet’s unit theorem |
---|---|

Canonical name | CharacterizingCMfieldsUsingDirichletsUnitTheorem |

Date of creation | 2013-03-22 17:57:26 |

Last modified on | 2013-03-22 17:57:26 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 4 |

Author | rm50 (10146) |

Entry type | Theorem |

Classification | msc 11R04 |

Classification | msc 11R27 |

Classification | msc 12D99 |