Dependencies

The additive character \(\psi : \mathcal{O}_K/P \to \mu _p(\mathcal{O}_L)\) is defined by

where the exponent is lifted canonically to \(\mathbb {Z}\).

For \(a \in \mathbb {Z}\), the Gauss sum is

For \(a \in \mathbb {Z}\), we define

The Teichmüller character is the unique group isomorphism

satisfying \(\omega (x) \equiv x \pmod{P}\) for all \(x \in (\mathcal{O}_K/P)^\times \), extended to all of \(\mathcal{O}_K/P\) by \(\omega (0) = 0\).

For all \(x \in \mathcal{O}_K/P\), \(\psi (x^p) = \psi (x)\).

Let \(\mathfrak {P}\) be a prime ideal of \(\mathcal{O}_L\) with \(\zeta _p - 1 \in \mathfrak {P}\). Then \(\psi (x) \equiv 1 \pmod{\mathfrak {P}}\) for all \(x \in \mathcal{O}_K/P\).

Let \(\mathfrak {P}\) be a prime ideal of \(\mathcal{O}_L\) with \(\zeta _p - 1 \in \mathfrak {P}\). Then

for all \(x \in \mathcal{O}_K/P\).

The character \(\psi \) is nontrivial, i.e., \(\psi \neq 1\).

The character \(\psi \) is primitive.

The extension \(L/F\) is Galois with \(\mathrm{Gal}(L/F) \cong (\mathbb {Z}/(q-1)\mathbb {Z})^\times \), via the action on \(\zeta _{q-1}\).

For all \(\sigma \in \mathrm{Gal}(L/F)\),

For all \(a \in \mathbb {Z}\) with \((q-1) \nmid a\),

For any prime \(\mathfrak {p}_0\) of \(\mathbb {Q}(\zeta _m)\) lying above \(p\) with \(\mathfrak {p}_0 \nmid m\),

as ideals of \(\mathcal{O}_{\mathbb {Q}(\zeta _m)}\).

For any prime \(\mathfrak {p}_0\) of \(\mathbb {Q}(\zeta _m)\) lying above \(p\) with \(\mathfrak {p}_0 \nmid m\),

as ideals of \(\mathcal{O}_{\mathbb {Q}(\zeta _m)}\).

For all \(a \in \mathbb {Z}\), \(g_{pa} = g_a\).

For any \(\tau \in \mathrm{Gal}(L/\mathbb {Q})\) fixing \(\zeta _p\),

For any \(\tau \in \mathrm{Gal}(L/K)\) with \(\tau (\zeta _p) = \zeta _p^e\),

If \((q-1) \nmid a\), then \(g_a \in \mathfrak {P}\).

If \((q-1) \nmid (a+b)\), then

If \((q-1) \nmid a\), then

If \((q-1) \nmid a\), then \(g_a \neq 0\).

If \((q-1) \nmid a\), then there exists \(k \geq 1\) such that \(N_{L/\mathbb {Q}}(g_a) = \pm p^k\). In particular, the only rational prime dividing \(N_{L/\mathbb {Q}}(g_a)\) is \(p\).

For any additive character \(\psi \) of \(\mathcal{O}_K/P\),

where \(\psi = 1\) denotes the trivial character (identically equal to \(1\)).

Assume \(\zeta _p - 1 \in \mathfrak {P}\). Then

If \((q-1) \mid k\), then for all \(a \in \mathbb {Z}\), \(g_{a+k} = g_a\).

For any \(\tau \in \mathrm{Gal}(L/\mathbb {Q}(\zeta _m, \zeta _p))\), \(\tau (g(\chi )^m) = g(\chi )^m\).

For any \(\tau \in \mathrm{Gal}(L/K)\), \(\tau (g(\chi )^m) = g(\chi )^m\).

We have \(g(\chi )^m \in \mathcal{O}_{\mathbb {Q}(\zeta _m)}\).

For all \(a, b \in \mathbb {Z}\), \(J(\omega ^{-a}, \omega ^{-b}) \in \mathcal{O}_K\).

If \((q-1) \nmid a\), \((q-1) \nmid b\), and \((q-1) \nmid (a+b)\), then \(J(\omega ^{-a}, \omega ^{-b}) \neq 0\).

Let \(F\) be a number field containing \(\mu _p\), let \(\mu \in F^\times \), and let \(L = F(\sqrt[p]{\mu })\). Let \(G_0 \leq \mathrm{Gal}(F/\mathbb {Q})\) be a subgroup. Then \(L/\mathbb {Q}\) is Galois with \(\mathrm{Gal}(L/\mathbb {Q})\) extending \(G_0\) if and only if for every \(\sigma \in G_0\) there exists \(\xi \in F^\times \) such that \(\sigma (\mu ) = \xi ^p \mu ^{a(\sigma )}\) for some \(a(\sigma ) \in \mathbb {Z}\). In particular, \(L/\mathbb {Q}\) is abelian if and only if for every \(\sigma _a \in \mathrm{Gal}(F/\mathbb {Q})\) there exists \(\xi \in F^\times \) such that \(\sigma _a(\mu ) = \xi ^p \mu ^a\).

Every finite abelian extension \(K/\mathbb {Q}\) is the compositum of cyclic extensions of prime power degree.

The Kummer extension \(L = F(\sqrt[p]{\mu })\) is abelian over \(\mathbb {Q}\) if and only if for every \(\sigma _a \in \mathrm{Gal}(F/\mathbb {Q})\) there exists \(\xi \in F^\times \) such that \(\sigma _a(\mu ) = \xi ^p \mu ^a\).

The ideal class \([\mathfrak {a}]\) is trivial, i.e., \(\mathfrak {a}\) is principal.

If \(K\) and \(K'\) are two cyclic \(p\)-extensions of \(\mathbb {Q}\) and their compositum \(KK'\) is cyclic, then \(K \subseteq K'\) or \(K' \subseteq K\).

With the above notation, \(L = F(\sqrt[p]{\mu })\) for some nonzero \(\mu \in \mathcal{O}_F\), and \(L/F\) is unramified outside \(p\).

Every extension of \(\mathbb {Q}\) of degree greater than \(1\) is ramified at some finite prime.

With the above notation, \((1 - \zeta _p) \nmid \mu \), and \((\mu ) = \mathfrak {a}^p\) for some ideal \(\mathfrak {a}\) of \(\mathcal{O}_F\).

There exists a unit \(\eta \in \mathcal{O}_F^\times \) such that \(\mu = \alpha ^p \eta \) for some \(\alpha \in \mathcal{O}_F\).

For any prime \(\mathfrak {q}\) of \(\mathcal{O}_F\), \(p \mid v_{\mathfrak {q}}(\mu )\).

Let \(K/\mathbb {Q}\) be a cyclic extension of prime power degree \(p^m\). If \(q \neq p\) is ramified in \(K/\mathbb {Q}\), then there exists a cyclic cyclotomic extension \(L/\mathbb {Q}\) such that \(KL = FL\) for some cyclic extension \(F/\mathbb {Q}\) of prime power degree in which \(q\) is unramified.

It suffices to prove that every cyclic extension of \(\mathbb {Q}\) of prime power degree unramified outside \(p\) is cyclotomic.

Let \(K/\mathbb {Q}\) be a cyclic extension of prime power degree \(p^m\). Then there exists a cyclic extension \(F/\mathbb {Q}\) of prime power degree \(p^m\) unramified outside \(p\) such that \(K\) is cyclotomic if and only if \(F\) is cyclotomic.

Let \(\mathfrak {q}\) be a prime ideal of \(F\) with \((\mu ) = \mathfrak {q}^r \mathfrak {a}\) with \(\mathfrak {q} \nmid \mathfrak {a}\). If \(p \nmid r\) and \(L/\mathbb {Q}\) is abelian, then \(\mathfrak {q}\) splits completely in \(F/\mathbb {Q}\).

The unit \(\eta \) is a \(p\)-th power times a root of unity, so \(\mu = \zeta _p^t\) for some \(t\). Therefore \(L = \mathbb {Q}(\zeta _{p^2})\).

Every prime ideal of \(\mathcal{O}_L\) lying above \(p\) is of the form \(\sigma (\mathfrak {P})\) for some \(\sigma \in \mathrm{Gal}(L/F)\). Moreover, \(\sigma (\mathfrak {P}) = \sigma '(\mathfrak {P})\) if and only if \(\sigma ^{-1}\sigma ' \in \langle p \bmod (q-1) \rangle \). In particular, each prime above \(p\) appears exactly \(f\) times in the family \(\{ \sigma (\mathfrak {P}) \mid \sigma \in \mathrm{Gal}(L/F)\} \).

The ramification index of \(\mathfrak {P}\) above \(p\) is \(p - 1\), i.e.,

For all \(a, b \in \mathbb {Z}\), there exists \(k \geq 0\) such that

For all \(a \geq 0\), there exists \(k \geq 0\) such that \(a = s(a) + k(p-1)\).

If \((q - 1) \mid a\), then \(s(a) = 0\).

For \(0 \leq h {\lt} q - 1\),

For all \(a \in \mathbb {Z}\), \(s(pa) = s(a)\).

For all \(a \in \mathbb {Z}\) with \(a \geq 0\), \(s(a) \leq a\).

Let \(0 \leq a\) and let \(a = a_0 + a_1 p + \cdots + a_{t-1} p^{t-1}\) be the \(p\)-adic expansion of \(a\). Then

For all \(a \in \mathbb {Z}\) with \(0 \leq a \leq p - 2\), \(s(a) = a\).

\(s(1) = 1\).

Assume \(f \geq 2\). Then \(s(p - 1) = p - 1\).

If \((q-1) \nmid a\), then \(s(a) {\gt} 0\).

For all \(a, b \in \mathbb {Z}\), \(s(a + b) \leq s(a) + s(b)\).

We have

Assume \((q-1) \mid k\). Then

The stabilizer of \(\mathfrak {P}\) in \(\mathrm{Gal}(L/F) \cong (\mathbb {Z}/(q-1)\mathbb {Z})^\times \) is the subgroup \(\langle p \bmod (q-1) \rangle \), which has order \(f\).

Let \(\sigma \) be a ring endomorphism of \(\mathcal{O}_L\) such that \(\sigma (\zeta _{q-1}) = \zeta _{q-1}^m\) for some \(m \in \mathbb {Z}\). Then \(\sigma (\omega (x)) = \omega (x)^m\) for all \(x \in \mathcal{O}_K/P\).

The order of \(\omega \) as a multiplicative character is \(q - 1\).

For all \(a \in \mathbb {Z}\) with \((q-1) \nmid a\),

There is a unique prime ideal of \(\mathcal{O}_F\) lying above \(p\), namely \(\mathfrak {P}\cap \mathcal{O}_F\).

If \(\mathbb {Q}(\zeta _m) \subseteq K \subseteq \mathbb {Q}(\zeta _m, \zeta _p)\) and \(K/\mathbb {Q}(\zeta _m)\) is unramified at all primes, then \(K = \mathbb {Q}(\zeta _m)\).

Any abelian extension of \(\mathbb {Q}(\zeta _m)\) contained in \(\mathbb {Q}(\zeta _m, \zeta _p)\) and unramified at all primes is trivial.

For all \(\sigma \in \mathrm{Gal}(L/F)\),

We have

In particular, \(v_{\mathfrak {P}}(J(\omega ^{-a}, \omega ^{-b})) \equiv 0 \pmod{p-1}\) (see Lemma 76 in the Appendix).

We have \(\zeta _p - 1 \notin \mathfrak {P}^2\).

Let \(K/\mathbb {Q}\) be a cyclic extension of degree \(2^m\) unramified outside \(2\). Then \(K\) is cyclotomic.

The only quadratic extensions of \(\mathbb {Q}\) unramified outside \(2\) are \(\mathbb {Q}(i)\), \(\mathbb {Q}(\sqrt{-2})\), and \(\mathbb {Q}(\sqrt{2})\). In particular, the maximal real abelian \(2\)-extension of \(\mathbb {Q}\) with exponent \(2\) and unramified outside \(2\) is \(\mathbb {Q}(\sqrt{2})\), which is the subfield of \(\mathbb {Q}(\zeta _8)\) of degree \(2\).

Let \(p\) be an odd prime. The maximal abelian extension of \(\mathbb {Q}\) of exponent \(p\) unramified outside \(p\) is cyclic of degree \(p\): it is the subfield of degree \(p\) of \(\mathbb {Q}(\zeta _{p^2})\).

Let \(K/\mathbb {Q}\) be a cyclic extension of odd prime power degree \(p^m\) unramified outside \(p\). Then \(K\) is cyclotomic.

Let \(0 \leq a {\lt} q - 1\) and let \(a = a_0 + a_1 p + \cdots + a_{f-1} p^{f-1}\), \(0 \leq a_i \leq p - 1\), be the \(p\)-adic expansion of \(a\). Then

Every abelian extension of \(\mathbb {Q}\) is contained in a cyclotomic field.

Let \(\mathfrak {p}_0\) be a prime of \(\mathbb {Q}(\zeta _m)\) with \(\mathfrak {p}_0 \nmid m\). Then \(\mathfrak {p}_0^{\theta }\) is principal in \(\mathbb {Q}(\zeta _m)\).