Algebraic number theory

Algorithm for finding integral basis

(1) Let w be any Q-basis for K s.t. w ⊆ OK. Calculate ∆(w)². Let M⊆OK be its Z-span. (2) If [OK : M] = m, then |∆(M)²| = m²|∆(OK)²|. If ∆(M)² is squarefree then m = 1 and OK = M. Otherwise, there exists p prime with p²|∆(M)² and c1,...,cn ∈ Z, not all divisible by p, such that 1/pΣcjwj ∈ OK. (3) if ∆(M)² is not squarefree than for each prime p such that p²|∆(M)², look for α∈OK of the form described. Suppose that p does not divide cj for j = k. Multiplying through by r ∈ Z such that rck ≡ 1 mod p, we may assume that ck ≡ 1 mod p. Subtracting integer multiples of the wi we may assume that 0≤ci<p for all i, and so ck =1. Replacing wk by our new α we get another basis, spanning a Z-module M′ with ∆(M′)² = 1/p²∆(M)² (4) Repeat the whole process with M′ instead of M. If α does not exist then p cannot divide m. Eventually we reach a basis for which none of the available primes divide m, so that m = 1 and we have arrived at an integral basis

Class numbers and PIDS

We have hK = 1 if and only if OK is a PID.

Alternative form for discriminant

We have ∆(w)2 = det(TrK/Q(wiwj)), and so ∆(w)^2 ∈ Q.

Convex subset

We say S ⊆ R^n is convex if x, y ∈ S, 0 ≤ λ ≤ 1 implies λx + (1−λ)y ∈ S.

Symmetric subset

We say S ⊆ R^n is symmetric (about the origin) if x ∈ S ⇒ −x ∈ S.

Coprime ideals

We say that A, B are coprime if A + B = OK

Integral basis

We say that w1, ..., wn ∈ OK is an integral basis for OK if OK ={Σcjwj :cj ∈Z}.

Set of algebraic integers

We say that α ∈ K is an algebraic integer if and only if there exists a monic g(x) ∈ Z[x] such that g(α) = 0. Define OK as the set of all algebraic integers in K.

Absolute conjugate, trace and norm

When K = Q(α), σi(α), TrK/Q(α), ΝormΚ/Q(α) are called the absolute conjugates, trace and norm.

Number field

A number field (or algebraic number field) is a finite extension K of Q. The index [K : Q] is the degree of the number field.

Unimodular matrix

A square matrix over Z is unimodular if it has determinant ±1.

Ideal classes and Minkowski's constant

Any ideal class c ∈ CK contains an ideal J such that N(J) ≤ cK

EDs, PIDs and UFDs

ED implies PID implies UFD. None of the reverse directions hold generally

Ideal class

Equivalence classes in OK under ∼ are called ideal classes. Let CK denote the set of ideal classes. The cardinality hK = |CK | is the class number of K.

Number field whose ring of integers is not a UFD

For K = Q(√−5) the ring OK = Z[√−5] is not a UFD.

Minkowski's constant

For a number field K, cK:= (4/π)^s*(n!/n^n)*√|∆^2(K)| is Minkowski's constant for K

Embeddings and the minimal polynomial

For any K = Q(β), suppose that β has minimal polynomial mβ(X). If β1, ..., βn are the n roots of mβ in C then one can choose the embeddings so that σi :β |→ βi.

Principal power of an ideal

For any nonzero ideal I ⊆ OK, there exists k such that 1≤k≤hK and I^k is principal.

Legendre symbol

For prime p and p does not divide m, define the Legendre symbol by (m/p) = 1 if m is a quadratic residue mod p and -1 otherwise. When p|m, we normally define (m/p) = 0

Special subset of R^n

For t > 0 let Rt := {(x1, ..., xr, z1, ..., zs) ∈ R^r×C^s : Σ|xi|+2Σ|zi| ≤ t}. Then 1. Rt is a compact, symmetric, and convex subset of R^n. 2. Vol(Rt) = 2^r*t^n*(π/2)^s/n!

Norm of a prime factorisation

If A = ΠPi^ei, (Pi being distinct nonzero prime ideals), then we have N(A) = ΠN(Pi)^ei .

Multiplication of coprime ideals

If A and B are coprime then AB = A ∩ B

Multiplicativity of norms

If A, B are nonzero ideals then N(AB) = N(A)N(B).

Norm of a principal ideal

If I = (α) then N(I) = |NormK/Q(α)|

Ideal sufficient criterion for algebraic integers

If I is a nonzero ideal of OK, and w ∈ K with wI ⊆ I, then w ∈ OK.

Cancellation for a principal ideal

If I, J are nonzero ideals in OK, and w ∈ OK is such that (w)I = JI, then (w) = J.

Cancellation for OK

If I, J ⊆ OK are ideals, with I nonzero, and JI = I then J = OK.

Properties of absolue trace and norm

If K = Q(α) and mα(X) = Xn +cn−1Xn−1 +···+c0, then we have TrK/Q(α) = −c[n−1] and NormK/Q(α) = (−1)^n*c0. In particular the norm and trace are in Q. More generally, for any K = Q(β), α ∈ K, the norm and trace of α are symmetric functions of the conjugates σi(α), and are therefore in Q.

Discriminant for number field with standard basis

If K = Q(α) and v = {1,α,...,αn−1} then ∆(v)2 = Π[i<j](αj − αi)^2. Here α1, ..., αn are the conjugates of α.

Primitive element theorem

If K is a number field then K = Q(θ) for some (algebraic) number θ ∈ K.

Discriminant of a number field

If K ≠ Q, then |∆^2(K)| > 1.

Size of quotients of prime ideals

If P is a non-zero prime ideal of OK and i ≥ 0 then #P^i/P^(i+1) = #OK/P.

Norm of a power of a prime

If P is a nonzero prime ideal and e ≥ 1 then N(P^e) = N(P)§e.

Multiplicativity of norms of coprime ideals

If nonzero A,B are coprime then N(AB) = N(A)N(B).

Fermat/Euler theorem (squares and primes)

If p is a prime, and p ≡ 1 mod 4, then there exist a, b ∈ Z such that p = a2 + b2, and this decomposition is unique. [here 'unique' means up to ± and up to swapping a and b.]

Ramification and discriminants

If p ramifies then p|∆(Z[α])^2.

Relating determinants

If v = {v1, ..., vn} is a basis for K/Q and w = {w1, ..., wn} ⊆ K, with wi = Σcijvj and cij ∈Q, then ∆(w) = det(C)∆(v) where C = (cij).

Trace and norm of algebraic integers

If α ∈ OK then TrK/Q(α), NormK/Q(α) ∈ Z.

Unique Factorisation Theorem for ideals of OK

Let A be any nonzero proper ideal of OK. Then there exist prime ideals P1, ..., Pr such that A = P1...Pr . The factorisation is unique up to the order of the factors; that is, if A = Q1 . . . Qs is another prime ideal factorisation then s=r and there exists a permutation σ such that Qi =Pσ(i), 1≤i≤r.

Addition of ideals

Let A, B be ideals. We define A+B := {a+b : a∈A,b∈B}, which is clearly an ideal.

To contain is to divide

Let A, B be nonzero ideals in OK. Then B ⊇ A if and only if there exists an ideal C such that A = BC, i.e., B|A.

Prime ideals and division

Let A, B be nonzero ideals, and P a prime ideal of OK such that P|AB. Then either P|A or P|B

Ideal division

Let A, B ⊆ OK be nonzero ideals. We write B|A if there exists an ideal C⊆OK such that A=BC.

Cancellation lemma

Let A, B, C ⊆ OK be nonzero ideals with AB=AC. Then B=C.

Size of quotient group

Let G be a free abelian group of rank n, and H a subgroup. Then G/H is finite if and only if H has rank n. Moreover, if G and H have Z-bases x1, ..., xn and y1, ..., yn with yi = Σaijxj we have #G/H = | det(aij )|.

Intersection of ideals and Z

Let I ⊆ OK be a non-zero ideal. Then I∩Z|≠ {0}.

Size of quotients of ring of integers

Let I ⊆ OK be a nonzero ideal. Then OK/I is a finite ring.

Minkowski's constant theorem

Let I ⊆ OK be a nonzero ideal. Then there exists a nonzero α∈I with |NormK/Q(α)| ≤ cK*N(I)

Embeddings of number fields

Let K = Q(θ) be a number field of degree n over Q. Then there are exactly n distinct monomorphisms (embeddings) σi : K → C (i = 1, ..., n). The elements σi(θ) are the distinct zeros in C of the minimal polynomial mθ of θ over Q.

Z-bases for quadratic fields

Let K = Q(√d), where d ∈ Z, d ≠ 1, with d squarefree. OK = <1, √d> if d = 2, 3 mod 4; OK = <1, (1+√d)/2> if d = 1 mod 4

Prime ideals lying above p

Let K be a number field of degree [K : Q] = n. Let P be a non-zero prime ideal of OK. Then P∩Z is a prime ideal of Z, and so is of the form pZ for some rational prime p. We therefore have P ⊇ pOK = (p). We say that P lies above the prime p. Suppose that (p) = P1^e1...Pr^er where P1, ..., Pr are distinct prime ideals in OK. Then P1, ..., Pr are the prime ideals lying above the rational prime p.

Hurwitz's lemma

Let K be a number field with [K : Q] = n. Then there exists a positive integer M, depending only on the choice of integral basis for OK, such that for any γ ∈ K, there exist w ∈ OK and1 ≤ t ≤ M, t ∈ Z with |NormK/Q(tγ − w)| < 1.

Ring of integers

Let K be an algebraic number field. If α,β ∈ OK then α + β, αβ ∈ OK. Hence OK is a ring, called the ring of integers of K.

Ideals in field/ring extensions

Let K, L be fields with K ⊆ L. Let I be an ideal of OK. Then I · OL is defined to be the ideal of OL generated by products of the form il, such that i ∈ I, l ∈ OL (sometimes called the image of I in OL). Note that, for any ideals I, J of OK, any n ∈ N and any principal ideal (a) = aOK of OK, (IJ)·OL = (I ·OL)(J ·OL), I^n ·OL = (I ·OL)^n and (a)·OL = aOL, the principal ideal of OL generated by the same element

Conjugate

Let K/Q be an algebraic number field of degree n, and let α ∈ K. Let σi : K → C be the n embeddings, i=1, ..., n. The σi(α) are called the (K-)conjugates of α.

Minkowski's Convex Body Theorem

Let L be a lattice in R^n. Let S be a bounded measurable subset of R^n which is convex and symmetric. If Vol(S) > 2^n*Vol(L) then there exists v ∈ L\{0} with v ∈ S. If S is closed, and therefore compact, then it is enough to have Vol(S) ≥ 2^n*Vol(L).

Blichfeldt's lemma

Let L be a lattice in Rn, and let S be a bounded, measurable subset of Rn such that Vol(S) > Vol(L). Then there exist x, y ∈ S with x ≠ y and such that x − y ∈ L.

Discriminant of a subset of OK

Let M be any subset of OK which has a Z-basis. Define ∆(M)2 := ∆(w)2 for any Z-basis w of M.

Norms in a ring of integers

Let OK be the ring of integers in a number field K, and α, β ∈ OK. Then 1. α is a unit (in OK) if and only if NormK/Q(α) = ±1. 2. If α and β are associates (in OK) then NormK/Q(α) = ±NormK/Q(β). 3. If NormK/Q(α) is a rational prime, i.e. a prime number in Z, then α is irreducible in OK.

Ideal multiplication

Let R be an integral domain, and let I, J be ideals of R. Then IJ :={Σaibi : ai ∈ I, bi ∈ J, k ≥ 1}

Prime ideal

Let R be an integral domain. An ideal I of R is prime if it is proper and (ab ∈ I ⇒ a ∈ I or b ∈ I). (recall: an ideal I of R is proper if I ≠ R).

Euclidean domain

Let R be an integral domain. R is a Euclidean domain (ED) if and only if there exists a function (a Euclidean function) d : R\{0} → N ∪ {0} such that 1. For all a, b ∈ R with b ≠ 0, there exist q, r ∈ R such that a = qb + r and either r = 0 or d(r) < d(b). 2. For all nonzero a, b ∈ R, d(a) ≤ d(ab).

Eisenstein

Let f(t) = a0 +a1t+···+ant^n ∈ Z[t]. Suppose there exists a prime p such that p does not divide an, but p divides ai for i = 0, ..., n−1, and p2 does not divide a0. Then, apart from constant factors, f(t) is irreducible over Z, and hence irreducible over Q.

Legendre symbol lemmas

Let p be an odd prime and let p not divide m, n, m1, m2. (a) If m1≡m2 (mod p) then (m1/p) = (m2/p). (b) (mn/p) = (m/p)(n/p). (c) (−1/p) =1 iff p≡1 (mod 4) or p=2. (−1/p) =−1 iff p≡3 (mod 4). (d) (2/p) =1 iff p≡±1 (mod 8). (2/p) = -1 iff p≡±3 (mod 8).

Quadratic residue

Let p be prime and m ∈ Z. We say that m is a quadratic residue mod p if there exists x ∈ Z such that m ≡ x^2 (mod p). Otherwise m is a quadratic non-residue mod p.

Gauss's Lemma

Let p(t) ∈ Z[t] be irreducible in Z[t]; then it is also irreducible in Q[t].

Determinant and discriminant

Let w = {w1, ..., wn} be an n-tuple of elements of K, where n = [K : Q]. The determinant is ∆(w) := det(σi(wj)), i.e., the determinant of the n×n matrix whose (i,j)th entry is σi(wj). The discriminant of w is ∆(w)^2. [sometimes also written as ∆^2(w).]

Q-basis which is not a Z-basis

Let w = {w1, ..., wn} be any Q-basis for K such that w ⊆ OK. Let M = ⟨w1, ..., wn⟩Z and let M ≠ OK. Then there exists p prime with p^2|∆(M)^2 and c1, ..., cn ∈ Z, not all divisible by p, such that 1/p(c1w1 + ... + cnwn) ∈ OK .

Fundamental domain

Let {v1, ..., vn} be any basis for R^n. Let D = {Σaivi : ai ∈ [0, 1)}. We call D a fundamental domain for the lattice L. Every v ∈ R^n can be expressed uniquely as v=u+w with u∈L and w∈D.

Lattice

Let {v1, ..., vn} be any basis for R^n. Let L = {Σaivi : ai ∈ Z} be the lattice generated by the vi. It is an additive subgroup of Rn.

Norm

NormK/Q(α) = NK/Q(α) = N(α) = Πσi(α).

Properties of the norm

NormK/Q(γδ) = NormK/Q(γ)*NormK/Q(δ); NormK/Q(γ) = 0 iff γ = 0; NormK/Q(q) = q^n for q ∈ Q.

Prime and irreducible elements

Prime elements are irreducible. If R is an ID in which everything factorises into irreducibles, then irreducible elements are prime iff R is a UFD

Unique factorisation domain

R is a unique factorisation domain (UFD) if and only if for all nonzero and non-unit α ∈ R there exist irreducible β1, ..., βn ∈ R such that 1. α = β1 ... βn. 2. If α = γ1 ...γm with irreducible γi, then m = n and there exists a permutation σ of {1, ..., n} such that βi and γσ(i) are associates.

Degree of a prime ideal

Suppose that (p) = P1^e1...Pr^er where P1, ..., Pr are distinct prime ideals in OK. Taking norms we have p^n =N(P1)^e1 ...N(Pr)^er. Hence, each N(Pi) = p^fi and Σeifi = n. The integer fi is called the degree of Pi.

Ramification index

Suppose that (p) = P1^e1...Pr^er where P1, ..., Pr are distinct prime ideals in OK. The integer ei is called the ramification index of Pi. If ei >1 we say that Pi is ramified. If some ei >1 we say that p ramifies in K.

Dedekind's theorem

Suppose that K = Q(α) with α ∈ OK having minimal polynomial m(x) ∈ Z[x] of degree n. If p does not divide [OK : Z[α]] and m*(x) := m(x) mod p ∈ Fp[x] factorises as m*(x) = Πg*i(x)^ei with g*i distinct and irreducible, then 1. Pi = (p, gi(α)) is a prime ideal of OK (here gi(x) ∈ Z[x] is any polyno- mial such that gi(x) ≡ g*i(x) mod p). 2.The prime ideals Pi are distinct. 3. The degree of P is the degree of g*i 4.(p)=ΠPi^ei.

Size of class number

The class number hK = #CK is finite.

Class group

The ideal classes form a group CK. It is called the class group of K and its order is the class number hK.

Norm of an ideal

The norm of I is defined as N(I) := #OK/I.

Fermat/Euler theorem 2 (integer solutions)

The only integer solution of y^2 + 2 = x^3 are x=3, y=±5

Integral basis theorem

The ring OK has an integral basis (that is, a Z-basis).

Trace

TrK/Q(α) = Σσi(α)

inert prime

we say that p is inert if (p) is prime in OK

splitting prime

we say that p splits if p does not ramify and is not inert

Z-module criterion for algebraic integers

α ∈ K is an algebraic integer if and only if there exists a non-zero finitely generated Z-module M ⊆ K such that αM ⊆ M.

Norm of a unit

α ∈ OK is a unit if and only if NormK/Q(α) = ±1.

Unit

α ∈ OK is a unit if and only if α−1 ∈ OK.

Determinants and bases

∆(w1 ...,wn) ≠ 0 if and only if w1, ..., wn is a basis for K/Q.

Equivalence relation on ideals

f I, J are nonzero ideals of OK, we write I ∼ J (and say that I is equivalent to J) if there exist α, β ∈ OK\{0} such that I(α) = J(β).