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[parent] examples of prime ideal decomposition in number fields (Example)

Here we follow the notation of the entry on the decomposition group. See also this entry.

Example 1

Let $ K=\mathbb{Q}(\sqrt{-7})$; then $ \operatorname{Gal}(K/\mathbb{Q})=\{\operatorname{Id},\sigma\} \cong \mathbb{Z}/2\mathbb{Z}$, where $ \sigma$ is the complex conjugation map. Let $ \mathcal{O}_K$ be the ring of integers of $ K$. In this case:

$\displaystyle \mathcal{O}_K=\mathbb{Z}\left[\frac{1+\sqrt{-7}}{2}\right]$
The discriminant of this field is $ D_{K/\mathbb{Q}}=-7$. We look at the decomposition in prime ideals of some prime ideals in $ \mathbb{Z}$:
  1. The only prime ideal in $ \mathbb{Z}$ that ramifies is $ (7)$:
    $\displaystyle (7)\mathcal{O}_K=(\sqrt{-7})^2$
    and we have $ e=2,f=g=1$. Next we compute the decomposition and inertia groups from the definitions. Notice that both $ \operatorname{Id}, \sigma$ fix the ideal $ (\sqrt{-7})$. Thus:
    $\displaystyle D((\sqrt{-7})/(7))=\operatorname{Gal}(K/\mathbb{Q})$
    For the inertia group, notice that $ \sigma\equiv \operatorname{Id}\ \operatorname{mod}\ (\sqrt{-7})$. Hence:
    $\displaystyle T((\sqrt{-7})/(7))=\operatorname{Gal}(K/\mathbb{Q})$
    Also note that this is trivial if we use the properties of the fixed field of $ D((\sqrt{-7})/(7))$ and $ T((\sqrt{-7})/(7))$ (see the section on “decomposition of extensions” in the entry on decomposition group), and the fact that $ e\cdot f\cdot g=n$, where $ n$ is the degree of the extension ($ n=2$ in our case).
  2. The primes $ (5),(13)$ are inert, i.e. they are prime ideals in $ \mathcal{O}_K$. Thus $ e=1=g,f=2$. Obviously the conjugation map $ \sigma$ fixes the ideals $ (5),(13)$, so
    $\displaystyle D(5\mathcal{O}_K/(5))=\operatorname{Gal}(K/\mathbb{Q})=D(13\mathcal{O}_K/(13))$
    On the other hand $ \sigma(\sqrt{-7})\equiv-\sqrt{-7} \operatorname{mod} (5),(13)$, so $ \sigma\neq \operatorname{Id} \operatorname{mod} (5),(13)$ and
    $\displaystyle T(5\mathcal{O}_K/(5))=\{\operatorname{Id}\}=T(13\mathcal{O}_K/(13))$
  3. The primes $ (2),(29)$ are split:
    $\displaystyle 2\mathcal{O}_K={\left( 2,\frac{1+\sqrt{-7}}{2}\right)}{\left( 2,\frac{1-\sqrt{-7}}{2}\right)}=\mathcal{P}\cdot\mathcal{P'}$
    $\displaystyle 29\mathcal{O}_K=\left(29,14+\sqrt{-7}\right)\left(29,14-\sqrt{-7}\right)=\mathcal{R}\cdot\mathcal{R'}$
    so $ e=f=1,g=2$ and
    $\displaystyle D(\mathcal{P}/(2))=T(\mathcal{P}/(2))=\{\operatorname{Id}\}=D(\mathcal{R}/(29))=T(\mathcal{R}/(29))$

Example 2

Let $ \zeta_7=e^{\frac{2\pi i}{7}}$, i.e. a $ 7^{th}$-root of unity, and let $ L=\mathbb{Q}(\zeta_7)$. This is a cyclotomic extension of $ \mathbb{Q}$ with Galois group

$\displaystyle \operatorname{Gal}(L/\mathbb{Q})\cong \left(\mathbb{Z}/7\mathbb{Z}\right)^{\times}\cong \mathbb{Z}/6\mathbb{Z}$
Moreover
$\displaystyle \operatorname{Gal}(L/\mathbb{Q})=\{\sigma_a\colon L\to L \mid \si... ...a(\zeta_7)=\zeta_7^a,\quad a\in \left(\mathbb{Z}/7\mathbb{Z}\right)^{\times} \}$

Galois theory gives us the subfields of $ L$: $ \xymatrix@dr@C=1pc{ L=\mathbb{Q}(\zeta_7) \ar@{-}[r] \ar@{-}[d] & \mathbb{Q}(\zeta_7+\zeta_7^6) \ar@{-}[d] \ \mathbb{Q}(\sqrt{-7}) \ar@{-}[r] & \mathbb{Q}}$

The discriminant of the extension $ {L/\mathbb{Q}}$ is $ D_{L/\mathbb{Q}}=-7^5$. Let $ \mathcal{O}_L$ denote the ring of integers of $ L$, thus $ \mathcal{O}_L=\mathbb{Z}[\zeta_7]$. We use the results of this entry to find the decomposition of the primes $ 2,5,7,13,29$:

$ \xymatrix{ {L=\mathbb{Q}(\zeta_7)} \ar@{-}[d]^3 & {(1-\zeta_7)^6} \ar@{-}[d] &... ...-}[d] & (5) \ar@{-}[d]& (13) \ar@{-}[d]\ \mathbb{Q}& (7) & (2) & (5) & (13) }$

  1. The prime ideal $ 7\mathbb{Z}$ is totally ramified in $ L$, and the only prime ideal that ramifies:
    $\displaystyle 7\mathcal{O}_L=(1-\zeta_7)^6=\mathfrak{T}^6$
    Thus
    $\displaystyle e(\mathfrak{T}/(7))=6,\quad f(\mathfrak{T}/(7))=g(\mathfrak{T}/(7))=1$
    Note that, by the properties of the fixed fields of decomposition and inertia groups, we must have $ L^{T(\mathfrak{T}/(7))}=\mathbb{Q}=L^{D(\mathfrak{T}/(7))}$, thus, by Galois theory,
    $\displaystyle D(\mathfrak{T}/(7))=T(\mathfrak{T}/(7))=\operatorname{Gal}(L/\mathbb{Q})$
  2. The ideal $ 2\mathbb{Z}$ factors in $ K$ as above, $ 2\mathcal{O}_K=\mathcal{P}\cdot\mathcal{P'}$, and each of the prime ideals $ \mathcal{P},\mathcal{P'}$ remains inert from $ K$ to $ L$, i.e. $ \mathcal{P}\mathcal{O}_L=\mathfrak{P}$, a prime ideal of $ L$. Note also that the order of $ 2\ \operatorname{mod}\ 7$ is $ 3$, and since $ g$ is at least $ 2$, $ 2\cdot3=6$, so $ e$ must equal $ 1$ (recall that $ efg=n$):
    $\displaystyle e(\mathfrak{P}/(2))=1,\quad f(\mathfrak{P}/(2))=3,\quad g(\mathfrak{P}/(2))=2$
    Since $ e=1$, $ L^{T(\mathfrak{P}/(2))}=L$, and $ [L\colon L^{D(\mathfrak{P}/(2))}]=3$, so
    $\displaystyle D(\mathfrak{P}/(2))=<\sigma_2>\cong \mathbb{Z}/3\mathbb{Z},\quad T(\mathfrak{P}/(2))=\{\operatorname{Id}\}$
  3. The ideal $ (5)$ is inert, $ 5\mathcal{O}_L=\mathfrak{S}$ is prime and the order of $ 5$ modulo $ 7$ is $ 6$. Thus:
    $\displaystyle e(\mathfrak{S}/(5))=1,\quad f(\mathfrak{S}/(5))=6,\quad g(\mathfrak{S}/(5))=1$
    $\displaystyle D(\mathfrak{S}/(5))=\operatorname{Gal}(L/\mathbb{Q}),\quad T(\mathfrak{S}/(5))=\{\operatorname{Id}\}$
  4. The prime ideal $ 13\mathbb{Z}$ is inert in $ K$ but it splits in $ L$, $ 13\mathcal{O}_L=\mathfrak{Q}_1\cdot\mathfrak{Q}_2\cdot\mathfrak{Q}_3$, and $ 13\equiv 6\equiv -1\ \operatorname{mod}\ 7$, so the order of $ 13$ is $ 2$:
    $\displaystyle e(\mathfrak{Q}_i/(13))=1,\quad f(\mathfrak{Q}_i/(13))=2,\quad g(\mathfrak{Q}_i/(13))=3$
    $\displaystyle D(\mathfrak{Q}_i/(13))=<\sigma_6>\cong\mathbb{Z}/2\mathbb{Z},\quad T(\mathfrak{Q}_i/(13))=\{\operatorname{Id}\}$
  5. The prime ideal $ 29\mathbb{Z}$ is splits completely in $ L$,
    $\displaystyle 29\mathcal{O}_L=\mathfrak{R}_1\cdot\mathfrak{R}_2\cdot\mathfrak{R}_3\cdot\mathfrak{R'}_1\cdot\mathfrak{R'}_2\cdot\mathfrak{R'}_3$
    Also $ 29\equiv 1\ \operatorname{mod}\ 7$, so $ f=1$,
    $\displaystyle e(\mathfrak{R}_i/(29))=1,\quad f(\mathfrak{R}_i/(29)=1,\quad g(\mathfrak{R}_i/(29))=6$
    $\displaystyle D(\mathfrak{R}_i/(29))=T(\mathfrak{R}_i/(29))=\{\operatorname{Id}\}$



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See Also: decomposition group, discriminant, number field, prime ideal decomposition in quadratic extensions of $\mathbb{Q}$, prime ideal decomposition in cyclotomic extensions of $\mathbb{Q}$, examples of ramification of archimedean places

Keywords:  decomposition, inertia, prime ideal decomposition

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Cross-references: order, factors, totally ramified, subfields, Galois theory, Galois group, cyclotomic extension, unity, conjugation, inert, primes, extension, degree, section, fixed field, properties, ideal, fix, definitions, inertia groups, ramifies, prime ideals, decomposition, field, discriminant, ring of integers, map, complex conjugation, decomposition group
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This is version 9 of examples of prime ideal decomposition in number fields, born on 2003-08-20, modified 2003-08-27.
Object id is 4628, canonical name is ExamplesOfPrimeIdealDecompositionInNumberFields.
Accessed 3374 times total.

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AMS MSC: 11S15 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Ramification and extension theory)
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