Lecture Summary MP473, Number Theory IIIH/IVH

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,...,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)=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 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,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 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 x²+2=y³, 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=x²+y², 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 - ((a₁,...,a_m),(b₁,...,b_n)=(a₁,...,a_m,b₁,...,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_r,s=(4/π)^sn!/nⁿ, I_K is generated by the prime ideal factors of the primes p ≤ M_r,s√|D_K|, examples: ℚ(√d), d=-1,2,-19,-43,-67,-163.; More examples:ℚ(√-5), ℚ(ζ_p), p=5,7, ℚ(m^1/3), m=2,3,5,6,7, the h_Kth 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 B_n(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 M_r,s=(4/π)^sn!/nⁿ theorem, lower bound for |D_K|, lattice-theoretic proof of Lagrange's four-squares theorem.

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KRM 2nd July 2006