Outline of the second half of the course
Disclaimer: This is intended to give you an overview of the second half of the course, and many of
the statements are rather vague, being intended to evoke the precise statements given in the course.
It is not an exhaustive list of examinable material. With a few exceptions, noted below, anything we
covered in the lectures and exercises should be considered examinable.
Chapter IV: Groups
14. group, subgroup, cyclic group, hai, order of a, o(a)
15. generators, hSi, direct product G × H
16. cosets
Ha = Hb ⇐⇒ ab−1 ∈ H, and this gives an equivalence relation on the group.
17. Lagrange’s theorem: If G is finite and H is a subgroup then |H| divides |G|.
In particular, o(a) = |hai| divides |G| for a ∈ G, and if |G| is prime then G is
cyclic.
18. group isomorphisms
19. Fundamental theorem of finite Abelian groups: Any finite Abelian group is
a direct product of cyclic groups Zpk of prime power order, and this “factorisation”
as a direct product is unique (up to re-ordering).
20. Cayley’s theorem: Any finite group is isomorphic to a permutation group (in
fact, to a subgroup of S|G| ).
Chapter V: Group homomorphisms
21. group homomorphisms, kernel, normal subgroups
θ is one-to-one ⇐⇒ ker θ = {e}.
22. quotient groups, the natural homomorphism
“Kernel of a group homomorphism” is the same as “normal subgroup”.
23. Fundamental homomorphism theorem:
If θ : G → H is an onto group homomorphism and N = ker θ then H ≈ G/N .
Chapter VI: Rings
24. ring, commutative ring, unital ring, unit element, zero element, direct sum R × S
25. zero divisor, integral domain, subring
26. field, subfield
A finite integral domain is a field.
27. ring isomorphisms, characteristic
char D is either 0 or prime for an integral domain D.
char D = 0 =⇒ D contains a copy of Z;
char D = p, prime =⇒ D contains a copy of Zp .
1
Chapter VII: Familiar number systems
28. ordered integral domains, Dp , a < b
The characteristic of any ordered integral domain is zero.
29. well-ordered integral domains
Up to isomorphism, Z is the unique well-ordered integral domain.
The proofs of results in sections 30–33 are non-examinable. However, you should still
understand and be able to apply the definitions and results.
30. field of quotients of an integral domain, FD ; Q = FZ
Up to isomorphism, FD is the smallest field containing D.
31. ordered fields, completeness
Up to isomorphism, there’s only one complete ordered field, R.
32. the complex numbers, algebraically closed field
33. complex roots of unity
Computations with complex roots of unity
Chapter VIII: Polynomials
34. polynomials with coefficients in R, R[x], monic
If R is a (commutative/unital) ring then so is R[x].
35. substitution
The Division Algorithm in F [x]
The Remainder Theorem and the Factor Theorem.
36. irreducible polynomial, the gcd of polynomials in F [x]
The Euclidean Algorithm in F [x]
The Unique Factorisation Theorem in F [x]
37. [omitted]
Chapter IX: Quotient rings
38. ring homomorphism, ideal, principal ideal (a)
θ is one-to-one ⇐⇒ ker θ = {0}.
39. quotient rings
“Kernel of a ring homomorphism” is the same as “ideal”.
Fundamental homomorphism theorem:
If θ : R → S is an onto ring homomorphism and I = ker θ then S ≈ R/I.
40. All ideals of F [x] are principal ideals.
Description of F [x]/(a(x)).
C ≈ R[x]/(x2 + 1)
41. factorisation (VIII) and ideals (IX).
[Non-examinable]
2