- 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(θ
_{1},...,θ_{n}), 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)=x
^{2}-d. - Lecture 11
- Galois groups of f(x)=(x
^{2}-a)(x^{2}-b) and f(x)=x^{3}-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 A
_{n}, Galois groups of cubics, the sign of the discriminant of a ℚ-basis, O_{K}- the ring formed by the algebraic integers lying in K, integral bases exist, D_{K}- the discriminant of K, index of a ℚ-basis for K consisting of algebraic integers, index of an element in O_{K}. - Lecture 13
- A sufficient condition for 1, θ,...,θ
^{n-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,t^{2}, 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 O_{K}, the unit group U_{K}. - Lecture 14
- An integral basis and discriminant for the p-th cyclotomic field, irreducibles and associates in O
_{K}, N_{K}(α)=± p (p a prime) ⇒ α is irreducible, examples of irreducibles in the Gaussian integers, α is a unit of O_{K}⇔ N_{K}(α)=±1, units of imaginary quadratic fields. - Lecture 15
- Structure of U
_{K}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(U_{K}) (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(U
_{K}) is finite and hence cyclic, determination of tor(U_{K}) 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 x
^{2}+2=y^{3}, every irreducible divides exactly one rational prime if O_{K}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=x
^{2}+y^{2}, etc. - Lecture 19
- Ideals in O
_{K}, 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 O_{K}/I is finite, N_{K}(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, N_{K}(I) belongs to I, there are only finitely many ideals having a prescribed norm. - Lecture 20
- N
_{K}((a))=|N_{K}(a)|, if O_{K}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 O_{K}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 - ((a
_{1},...,a_{m}),(b_{1},...,b_{n})=(a_{1},...,a_{m},b_{1},...,b_{n}), 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, O_{K}is a UFD if and only if O_{K}is a PID, N_{K}(AB)=N_{K}(A)N_{K}(B). - Lecture 22
- If N
_{K}(P) is a rational prime, then P is a prime ideal, if P is a prime ideal dividing p, then N_{K}(P)=p^{f}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 I
_{K}and class number h_{K}, I_{K}is finite via the Minkowski constant*M*, I_{r,s}=(4/π)^{s}n!/n^{n}_{K}is generated by the prime ideal factors of the primes p ≤*M*, examples: ℚ(√d), d=-1,2,-19,-43,-67,-163._{r,s}√|D_{K}| - More examples:ℚ(√-5), ℚ(ζ
_{p}), p=5,7, ℚ(m^{1/3}), m=2,3,5,6,7, the h_{K}th power of an ideal is principal, gcd's exist in the ring of all algebraic integers. - Lecture 25
- Lattices in ℝ
^{n}, discrete subgroups of ℝ^{n}, volume (or determinant) of a lattice, a non-trivial discrete subgroup of ℝ^{n}is a lattice, volume (Jordan content), examples of parallelopiped and B_{n}(0,r). - Lecture 26
- Minkowski's V(E) > 2
^{n}theorem, compact form of Minkowski's theorem, application to p=x^{2}+y^{2}, proof of the earlier-mentioned*M*theorem, lower bound for |D_{r,s}=(4/π)^{s}n!/n^{n}_{K}|, lattice-theoretic proof of Lagrange's four-squares theorem.

KRM 2nd July 2006