Lecture Summary, MP473 Number Theory IIIH/IVH, Semester 2, 2000

Lecture 1
Algebraic numbers and integers.
Lecture 2
The algebraic numbers form a field, the algebraic integers form a ring.
Lecture 3
Proof of the two quadratic reciprocity laws using cyclotomy.
Lecture 4
Another type of Gaussian sum and Schur's proof of the sign of the Gaussian sum.
Lecture 5
The minimum polynomial of an algebraic number, Gauss' Lemma
Lecture 6
Algebraic number fields, [L : K], the field ℚ(θ), quadratic and cyclotomic fields
Lecture 7
Examples of cyclotomic polynomials, Heinz Lüneberg's account of cyclotomic polynomials, culminating in an algorithm for calculating the mth cyclotomic polynomial which is used in CMATR
Lecture 8
Irreducibility of the mth cyclotomic polynomial, prime divisors of the mth cyclotomic polynomial
Lecture 9
There are infinitely many primes in the congruence class 1 (mod m), the field ℚ(θ1,...,θn), the primitive element theorem, norm and trace
Lecture 10
The field polynomial is a power of the minimal polynomial, properties of norm and trace, splitting fields and normal extensions, K-isomorphisms and K-automorphisms, there are [L : K] K-isomorphims of L, a normal extension L has [L : ℚ] automorphisms, the Galois group of a normal extension and polynomial f(x) with rational coefficients, examples: f(x)=x2-d.
Lecture 11
Galois groups of f(x)=(x2-a)(x2-b) and f(x)=x3-d, the cyclotomic polynomial, ℚ is the fixed field of the set of ℚ-isomorphisms of K, the roots of the field polynomial of θ are the images of θ under the isomorphisms of K, discriminant of a field basis, non-vanishing of the discriminant, effect of change of basis on the discriminant, discriminant in terms of conjugates of basis elements, resultant R(f(x),g(x)) of two polynomials, R(f(x),g(x))=0 if and only if f(x) and g(x) have a non-trivial factor in common, Disc(f(x)) the discriminant of f(x).
Lecture 12
Various formulae for Disc(f(x)), conditions for the Galois group of f(x) to be a subgroup of An, Galois groups of cubics, the sign of the discriminant of a ℚ-basis, OK - the ring formed by the algebraic integers lying in K, integral bases exist, DK - the discriminant of K, index of a ℚ-basis for K consisting of algebraic integers, index of an element in OK.
Lecture 13
A sufficient condition for 1, θ,...,θn-1 to be an integral basis of ℚ(θ), integral bases of ℚ(√d) , a cubic example of Dedekind where there is no integral basis of the form 1,t,t2, a useful lemma on number fields defined by an algebraic integer having Eisensteinian minimum polynomial, application of this result to finding integral bases for pure cubic fields, divisibility in OK, the unit group UK.
Lecture 14
An integral basis and discriminant for the p-th cyclotomic field, irreducibles and associates in OK, NK(α)=± p (p a prime) ⇒ α is irreducible, examples of irreducibles in the Gaussian integers, α is a unit of OK ⇔ NK(α)=±1, units of imaginary quadratic fields.
Lecture 15
Structure of UK when K is a real quadratic field, statement of Dirichlet's t=r+s-1 unit theorem, the cyclotomic units sin(rπ/p)/sin(π/p) (r=2,...,(p-1)/2), tor(UK) (the group of units of finite order)={-1,1} if K possesses a real isomorphism.
Lecture 16
Kronecker's lemma on integers with conjugates not exceeding a given bound, tor(UK) is finite and hence cyclic, determination of tor(UK) when K is the pth cyclotomic field, every non-zero non-unit is a product of irreducibles, unique factorization domain (UFD), Euclidean number fields are UFD's.
Lecture 17
ℚ(√d) is norm-Euclidean if d=-1,-2,-3,-7,-11,2,3, or 5, gcd(14+13i,3-9i)=1+2i in the Gaussian integers, the diophantine equation x2+2=y3, every irreducible divides exactly one rational prime if OK is a UFD, splitting of rational primes into irreducibles.
Lecture 18
Decomposition of rational primes in ℚ(√d) , examples of d=-1,-3, 2 and applications to p=x2+y2, etc.
Lecture 19
Ideals in OK, nonzero ideals have ideal bases, which are all related by unimodular transformations, review of Hermite normal form, canonical ideal basis, there are only finitely many ideals containing a given nonzero rational integer, congruence modulo an ideal, quotient ring OK/I is finite, NK(I) (the norm of the ideal I) equals |det(B)|, where B is the matrix which arises on expressing a given ideal basis for I in terms of an integral basis, NK(I) belongs to I, there are only finitely many ideals having a prescribed norm.
Lecture 20
NK((a))=|NK(a)|, if OK is a UFD and p is irreducible, then (p) is a prime ideal, two quadratic field examples of prime ideals, product of two ideals, the non-zero ideals of OK form a cancellative semigroup, "to divide is to contain", every ideal has only finitely many ideal divisors.
Lecture 21
If A=BC and B != (1), then C has fewer ideal factors than A, if A !=(1) and is not a prime ideal, then A is a product of two non-trivial ideals, every ideal != (1) is a product of prime ideals, (A,B)= gcd of ideals A and B, (A,B) exists - ((a1,...,am),(b1,...,bn)=(a1,...,am,b1,...,bn), C(A,B)=(CA,CB), if P is a prime ideal and A|P, then A=P or A=(1), (P,A)=(1) if P does not divide A, P|AB implies P|A or P|B, factorisation into prime ideals is unique, every prime ideal divides exactly one rational prime, OK is a UFD if and only if OK is a PID, NK(AB)=NK(A)NK(B).
Lecture 22
If NK(P) is a rational prime, then P is a prime ideal, if P is a prime ideal dividing p, then NK(P)=pf and f=degree(P), the Kummer-Dedekind theorem describing the prime ideal decomposition of (p) for almost all p, examples: quadratic and cyclotomic fields.
Lecture 23
Equivalence of ideals, ideal classes, the ideal class group IK and class number hK, IK is finite via the Minkowski constant Mr,s=(4/π)sn!/nn, IK is generated by the prime ideal factors of the primes p ≤ Mr,s√|DK|, examples: ℚ(√d), d=-1,2,-19,-43,-67,-163.
More examples:ℚ(√-5), ℚ(ζp), p=5,7, ℚ(m1/3), m=2,3,5,6,7, the hKth power of an ideal is principal, gcd's exist in the ring of all algebraic integers.
Lecture 25
Lattices in ℝn, discrete subgroups of ℝn, volume (or determinant) of a lattice, a non-trivial discrete subgroup of ℝn is a lattice, volume (Jordan content), examples of parallelopiped and Bn(0,r).
Lecture 26
Minkowski's V(E) > 2n theorem, compact form of Minkowski's theorem, application to p=x2+y2, proof of the earlier-mentioned Mr,s=(4/π)sn!/nn theorem, lower bound for |DK|, lattice-theoretic proof of Lagrange's four-squares theorem.