Dependencies

Let \(p\) be an odd prime. Any two cyclic extensions \(K_1, K_2 \subseteq A\) of \(\mathbb {Q}\) of degree \(p\), unramified outside \(p\), are equal.

Let \(k\) be a number field isomorphic to \(\mathbb {Q}(\zeta _m)\). The Stickelberger element \(\Theta \) annihilates the ideal class group \(\mathrm{Cl}(k)\): for every ideal class \(c\),

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

We also view it as a character with values in \(\mathcal{O}_L\).

For each prime \(\mathfrak {q}\) of \(\mathcal{O}_k\) above \(p\), choose \(a_{\mathfrak {q}} \in (\mathbb {Z}/m\mathbb {Z})^\times \) with \(\sigma _{a_{\mathfrak {q}}}^{-1}(\mathfrak {p}) = \mathfrak {q}\). The map

is a bijection, and the image \(a\) of \((\mathfrak {q}, j)\) satisfies \(\sigma _a^{-1}(\mathfrak {p}) = \mathfrak {q}\).

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

Let \(R\) be a commutative ring, \(I\) an ideal of \(R\) and \(n\) a positive integer. The ideal \(I\) has Teichmüller lifts of order \(n\) if the reduction map \(\mu _n(R) \to (R/I)^\times \) is bijective.

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

For \(a \in \mathbb {Z}\) and a maximal ideal \(\mathfrak {Q}\) of \(\mathcal{O}_L\), let \(s_{\mathfrak {Q}}(a) \in \mathbb {N}\) be the multiplicity of \(\mathfrak {Q}\) in the ideal \((g_a)\). We write \(s(a) = s_{\mathfrak {P}}(a) = v_{\mathfrak {P}}(g_a)\).

An element \(\tau \in I(Q)\) acts trivially on \(S/Q\), so it acts on \(Q/Q^2\) by an \(S/Q\)-linear automorphism. The tame character is the determinant of this action,

It is a group homomorphism because the determinant is multiplicative.

Assume that \(P\) has Teichmüller lifts of order \(q - 1\) (2). The Teichmüller character \(\omega : \mathcal{O}_K/P \to \mathcal{O}_K\) is the multiplicative character whose restriction to \((\mathcal{O}_K/P)^\times \) is the inverse of the reduction map \(\mu _{q-1}(\mathcal{O}_K) \xrightarrow {\sim } (\mathcal{O}_K/P)^\times \), extended by \(\omega (0) = 0\).

Let \(p\) be a prime. A number field \(K\) is unramified outside \(p\) if every prime number \(q \neq p\) is unramified in \(K\).

Assume moreover that \(G\) is abelian and acts faithfully on \(S\), and that \(Q\) is tame. Then \(\# I(Q)\) divides \(N\mathfrak {p}- 1\).

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

The character \(\psi \) is primitive.

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

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

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

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

Let \(H_1, H_2\) be subgroups of a finite cyclic group \(C\) with \(H_1 \cap H_2 = 1\). Then \(\# C\) divides \(\operatorname {lcm}([C : H_1], [C : H_2])\).

Let \(R\) be a Dedekind domain and \(J\) a nonzero ideal of \(R\). Every ideal class of \(R\) contains an integral ideal coprime to \(J\).

For all \(\sigma \in \mathrm{Gal}(L/F)\) and \(a \in \mathbb {Z}\),

Under the isomorphism \(\mathrm{Gal}(\mathbb {Q}(\zeta _n)/\mathbb {Q}) \cong (\mathbb {Z}/n\mathbb {Z})^\times \), complex conjugation corresponds to \(-1\).

Restriction to \(K\) induces an isomorphism \(\mathrm{Gal}(L/F) \cong \mathrm{Gal}(K/\mathbb {Q}) \cong (\mathbb {Z}/(q-1)\mathbb {Z})^\times \). For \(\sigma \in \mathrm{Gal}(L/F)\), we denote by \(\mathfrak {n}(\sigma ) \in \{ 0, \ldots , q-2\} \) the representative of its image, so that \(\sigma (\zeta _{q-1}) = \zeta _{q-1}^{\mathfrak {n}(\sigma )}\).

Let \(\Gamma \in \mathcal{O}_k\) with \(g_d^m = \Gamma \) (50). Then

as ideals of \(\mathcal{O}_k\).

Let \(\tau \in \mathrm{Gal}(L/K)\) and \(e \in (\mathbb {Z}/p\mathbb {Z})^\times \) with \(\tau (\zeta _p) = \zeta _p^e\). Then for all \(a \in \mathbb {Z}\),

Let \(E = \mathbb {Q}(\zeta _{mp}) \subseteq L\). For any \(\tau \in \mathrm{Gal}(L/E)\), we have \(\tau (g_d) = g_d\).

For any \(\tau \in \mathrm{Gal}(L/K)\), we have \(\tau (g_d^m) = g_d^m\).

There exists \(\Gamma \in \mathcal{O}_k\) with \(g_d^m = \Gamma \).

For all \(a \in \mathbb {Z}\), we have the equality of ideals of \(\mathcal{O}_L\)

where the product runs over the primes of \(\mathcal{O}_L\) above \(p\).

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

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

If \((q-1) \nmid a\), then \(g_a \in (\zeta _p - 1)\mathcal{O}_L\).

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

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

For all \(a \in \mathbb {Z}\), we have \(g_a \neq 0\). If \((q-1) \nmid (a+b)\), then \(J(a, b) \neq 0\).

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

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

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

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

The ideal \(P\) has Teichmüller lifts of order \(q - 1\).

Let \(L/K\) be a finite Galois extension with group \(G\), let \(E\) be an intermediate field, and let \(I\) be a normal subgroup of \(G\) that is a complement of \(\mathrm{Gal}(L/E)\). Then the fixed field \(F = L^{I}\) satisfies \(FE = L\), and \(G/I \cong \mathrm{Gal}(L/E)\).

Let \(L/\mathbb {Q}\) be a finite abelian extension, and let \(K, E \subseteq L\) be subfields with \(KE = L\), \([K:\mathbb {Q}] = p^m\) and \(E \cong \mathbb {Q}(\zeta _q)\), where \(q \neq p\) are primes and \(q \nmid [L:\mathbb {Q}]\). Let \(\mathfrak {Q}\) be a prime of \(L\) above \(q\). Then the inertia group \(I = I(\mathfrak {Q}) \subseteq \mathrm{Gal}(L/\mathbb {Q})\) is a complement of \(\mathrm{Gal}(L/E)\): one has \(I \cap \mathrm{Gal}(L/E) = 1\) and \(\# I \cdot \# \mathrm{Gal}(L/E) = [L:\mathbb {Q}]\).

A quadratic field unramified outside \(2\) has discriminant \(-8\), \(-4\) or \(8\).

Every finite abelian extension \(K/F\) is the compositum of finitely many subextensions \(C_i/F\), each of which is cyclic of prime power degree.

For every \(a \in (\mathbb {Z}/p\mathbb {Z})^\times \), there is a nonzero \(\xi \in F\) with \(\sigma _a(\mu ) = \xi ^p \mu ^a\).

Let \(\mathfrak {a}\) be a nonzero ideal of \(\mathcal{O}_F\) with \(\mathfrak {a}^p = (\mu )\). Then \(\mathfrak {a}\) is principal.

Let \(\alpha \in \mathcal{O}_F\) and \(\eta \in \mathcal{O}_F^\times \) with \(\mu = \alpha ^p\eta \). Then \(\eta \, \sigma _{-1}(\eta )\) is a \(p\)-th power in \(F\).

Let \(K, K' \subseteq A\) be number fields such that \(KK'/\mathbb {Q}\) is Galois with cyclic Galois group. If \([K:\mathbb {Q}]\) divides \([K':\mathbb {Q}]\), then \(K \subseteq K'\).

Let \(p\) be a prime and \(K \subseteq A\) a cyclic extension of \(\mathbb {Q}\) of degree \(p^m\). Then \(K \subseteq \mathbb {Q}(\zeta _n)\) for some \(n \geq 1\).

Let \(K, K' \subseteq A\) be abelian extensions of \(\mathbb {Q}\) of the same degree \(p^m\), both unramified outside \(p\). Assume that any two cyclic subextensions of \(KK'\) of degree \(p\) over \(\mathbb {Q}\) and unramified outside \(p\) are equal. Then \(K = K'\).

There is a nonzero \(\alpha \in \mathcal{O}_L\) with \(\alpha ^p = \mu \).

Let \(\alpha \in \mathcal{O}_F\) and \(\eta \in \mathcal{O}_F^\times \) with \(\mu = \alpha ^p\eta \). Then \(\eta ^4 = \zeta \, \delta ^p\) for some \(p\)-th root of unity \(\zeta \) and some nonzero \(\delta \in F\).

Let \(C \subseteq A\) be an abelian extension of \(\mathbb {Q}\), unramified outside \(p\), and let \(H\) be a subgroup of \(\mathrm{Gal}(C/\mathbb {Q})\) of index \(p^m\) with cyclic quotient. Then the fixed field \(C^{H}\) is a cyclic extension of \(\mathbb {Q}\) of degree \(p^m\), unramified outside \(p\).

Let \(L/F\) be a cyclic extension of degree \(p\). Then there is a nonzero \(\mu \in \mathcal{O}_F\) such that \(L\) is the splitting field of \(X^p - \mu \) over \(F\).

Let \(L/F\) be a finite extension such that \(L/\mathbb {Q}\) is Galois and \([L:\mathbb {Q}] = p(p-1)\). Then \(L/F\) is Galois, cyclic of degree \(p\).

Let \(L \subseteq A\) be a CM field with complex conjugation \(c \in \mathrm{Gal}(L/\mathbb {Q})\). Then its maximal real subfield \(L^+\) is the fixed field of \(\langle c\rangle \).

Assume that \(L\) is unramified outside \(p\). Then \((\mu ) = \mathfrak {a}^p\) for some nonzero ideal \(\mathfrak {a}\) of \(\mathcal{O}_F\).

Let \(\mathfrak {a}\) be a nonzero principal ideal of \(\mathcal{O}_F\) with \(\mathfrak {a}^p = (\mu )\). Then \(\mu = \alpha ^p\eta \) for some nonzero \(\alpha \in \mathcal{O}_F\) and some unit \(\eta \in \mathcal{O}_F^\times \).

Assume that \(L\) is unramified outside \(p\). Then \(p \mid v_{\mathfrak {q}}(\mu )\) for every prime \(\mathfrak {q}\) of \(\mathcal{O}_F\).

Let \(K \subseteq A\) be a cyclic extension of \(\mathbb {Q}\) of degree \(p^m\), and let \(q \neq p\) be a prime that ramifies in \(K\). Then there is a cyclic extension \(F \subseteq A\) of \(\mathbb {Q}\) of \(p\)-power degree such that

• the prime \(q\) is unramified in \(F\);

• every prime that is unramified in \(K\) is unramified in \(F\);

• one has \(K\mathbb {Q}(\zeta _q) = F\mathbb {Q}(\zeta _q)\).

Let \(K \subseteq A\) be a number field with \(K/\mathbb {Q}\) abelian. Then \(K \subseteq \mathbb {Q}(\zeta _n)\) for some \(n \geq 1\).

Let \(K \subseteq A\) be a cyclic extension of \(\mathbb {Q}\) of degree \(p^m\). Then there exist a cyclic extension \(F \subseteq A\) of \(\mathbb {Q}\) of \(p\)-power degree, unramified outside \(p\), and an integer \(n \geq 1\) such that \(K\mathbb {Q}(\zeta _n) = F\mathbb {Q}(\zeta _n)\).

Let \(\mathfrak {q}\) be a prime of \(\mathcal{O}_F\) such that \(p \nmid v_{\mathfrak {q}}(\mu )\). Then every \(\sigma \in \mathrm{Gal}(F/\mathbb {Q})\) with \(\sigma (\mathfrak {q}) = \mathfrak {q}\) is the identity.

Let \(\mathfrak {a}\) be a nonzero ideal of \(\mathcal{O}_F\) with \(\mathfrak {a}^p = (\mu )\). Then \(L\) is the cyclotomic field of level \(p^2\), that is, \(L = \mathbb {Q}(\xi )\) for a primitive \(p^2\)-th root of unity \(\xi \in L\).

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

We have

For all \(x \in \mathcal{O}_K/P\), we have \(\omega (x) \equiv x \pmod{P}\).

If \((q-1) \nmid a\), then there exists \(k \geq 1\) such that \(|N_{L/\mathbb {Q}}(g_a)| = p^k\).

If \((q-1) \nmid a\), then \(g_a\) is not a unit of \(\mathcal{O}_L\).

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

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

Let \(\mathfrak {q}\) be a prime of \(\mathcal{O}_k\) above \(p\) and \(\mathfrak {Q}\) a prime of \(\mathcal{O}_L\) above \(\mathfrak {q}\). Then \(e(\mathfrak {Q}\mid \mathfrak {q}) = p - 1\).

Let \(K/\mathbb {Q}\) be a finite Galois extension, \(E, F\) two Galois subextensions with \(EF = K\), and \(\mathfrak {P}\) a prime of \(K\) above a prime number \(p\) with \(p \nmid [K:\mathbb {Q}]\). Then

Every prime ideal of \(\mathcal{O}_L\) above \(p\) is unramified over \(\mathcal{O}_F\).

Let \(m, d\) be positive integers with \(q - 1 = md\), and let \(a \in \mathbb {N}\) with \(da \leq q - 2\). Then

We have

The stabilizer of \(\mathfrak {p}\) in \(G\) is the subgroup \(\langle \sigma _p \rangle \), which has order \(f\).

In the setting of this chapter, the ideal \(\mathfrak {p}^{\Theta }\) of \(\mathcal{O}_k\) is principal.

Let \(b {\gt} 1\), \(l \geq 0\) and \(0 \leq x {\lt} b^l - 1\). Then

where \(S_b(x)\) is the sum of the base \(b\) digits of \(x\).

Let \(\sigma \in D(Q)\) be an arithmetic Frobenius, that is, \(\sigma (x) \equiv x^{N\mathfrak {p}} \pmod Q\) for all \(x \in S\). Then \(\theta (\sigma \tau \sigma ^{-1}) = \theta (\tau )^{N\mathfrak {p}}\) for all \(\tau \in I(Q)\).

Assume that \(G\) acts faithfully on \(S\). Let \(\tau \in I(Q)\) be such that the order \(m\) of \(\tau \) is not zero in \(S/Q\) and \(\theta (\tau ) = 1\). Then \(\tau = 1\).

For \(\tau \in I(Q)\) and \(v \in Q/Q^2\), one has \(\tau v = \theta (\tau ) v\).

Assume that \(G\) acts faithfully on \(S\) and that \(Q\) is tame. Then \(\theta \) is injective, and the inertia group \(I(Q)\) is cyclic.

Let \(a \in \mathbb {Z}\) with \((q-1) \nmid a\). The character \(\omega ^{-a}\), viewed with values in \(\mathcal{O}_L\), is nontrivial. In particular,

If \(K\) and \(F\) are number fields unramified outside \(p\), then so is \(KF\).

If \((q-1) \nmid (a+b)\), then \(s(a) + s(b) = s(a+b) + v_{\mathfrak {P}}(J(a, b))\).

If \((q-1) \nmid a\), then \(s(a) + s(-a) = f \cdot e(\mathfrak {P}\mid p)\).

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

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

Let \(\mathfrak {Q}\) be a maximal ideal of \(\mathcal{O}_L\) not lying above \(p\). Then \(s_{\mathfrak {Q}}(a) = 0\) for all \(a \in \mathbb {Z}\).

For all \(a \in \mathbb {N}\), \(s(a) \leq a\).

Let \(a \in \mathbb {N}\) and let \(a = a_0 + a_1 p + \cdots + a_{t-1} p^{t-1}\) be its \(p\)-adic expansion. Then

We have \(s(1) = 1\).

If \(a \equiv b \pmod{q-1}\), then \(s(a) = s(b)\).

For all \(\sigma \in \mathrm{Gal}(L/F)\) and \(a \in \mathbb {Z}\), \(s_{\sigma (\mathfrak {P})}(a) = s_{\mathfrak {P}}(a \cdot \mathfrak {n}(\sigma ^{-1}))\).

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

Let \(\mathfrak {q}\) be a prime of \(\mathcal{O}_k\) above \(p\) and \(\mathfrak {Q}\) a prime of \(\mathcal{O}_L\) above \(\mathfrak {q}\). Then

where \(b_j \in \{ 0, \ldots , m-1\} \) represents the image of \((\mathfrak {q}, j)\) under the bijection of 53.

Let \(\mathfrak {q}\) be a prime of \(\mathcal{O}_k\) above \(p\) and \(a \in (\mathbb {Z}/m\mathbb {Z})^\times \) with \(\sigma _a^{-1}(\mathfrak {p}) = \mathfrak {q}\). Then the sum of the (representatives of the) images of \((\mathfrak {q}, j)\), \(0 \leq j {\lt} f\), under the bijection of 53 is \(\sum _{j=0}^{f-1} (ap^j \bmod m)\). In particular, \(v_{\mathfrak {q}}(\mathfrak {p}^{\Theta }) = \sum _{j=0}^{f-1} (ap^j \bmod m)\).

For every prime ideal \(\mathfrak {Q}\) of \(\mathcal{O}_L\) above \(p\), we have \(\zeta _p - 1 \in \mathfrak {Q}\).

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

For every \(n \geq 0\), the quotient \((\mathbb {Z}/2^{n+2}\mathbb {Z})^\times / \langle -1 \rangle \) is cyclic.

Let \(K \subseteq A\) be a cyclic extension of \(\mathbb {Q}\) of degree \(2^m\), unramified outside \(2\). Then \(K \subseteq \mathbb {Q}(\zeta _{2^{m+2}})\).

Let \(K \subseteq A\) be a totally real cyclic extension of \(\mathbb {Q}\) of degree \(2^m\), unramified outside \(2\). Then \(K \subseteq \mathbb {Q}(\zeta _{2^{m+2}})\).

Any two totally real quadratic fields \(K_1, K_2 \subseteq A\) unramified outside \(2\) are equal.

Let \(p\) be an odd prime and \(K \subseteq A\) a cyclic extension of \(\mathbb {Q}\) of degree \(p\), unramified outside \(p\). Then \(K \subseteq \mathbb {Q}(\zeta _{p^2})\).

Let \(p\) be an odd prime, \(m \geq 1\), and \(K \subseteq A\) an abelian extension of \(\mathbb {Q}\) of degree \(p^m\), unramified outside \(p\). Then \(K \subseteq \mathbb {Q}(\zeta _{p^{m+1}})\).

Let \(0 \leq a \leq q - 2\) 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

Let \(K\) be a number field such that \(K/\mathbb {Q}\) is abelian. Then there exists \(n\) such that \(K\) embeds into the cyclotomic field \(\mathbb {Q}(\zeta _n)\).

Let \(k\) be a number field isomorphic to \(\mathbb {Q}(\zeta _m)\), let \(p\) be an odd prime with \(p \nmid m\) and let \(\mathfrak {p}\) be a maximal ideal of \(\mathcal{O}_k\) above \(p\). Then the ideal \(\mathfrak {p}^{\Theta }\) is principal.