Sections, concepts, and problems - Last updated April 19, 2005.

Section 14. Factor groups. Making the set of cosets of a normal subgroup into a group, the "fundamental homomorphism theorem", automorphisms, inner automorphisms, conjugate elements, conjugate subgroups.
7, 13, 20, 23, 27, 31, 34 (hint: what happens to H under every inner automorphism?), 40a

Section 15. Factor-group computations and simple groups. Playing around with factor groups (a.k.a. quotient groups), simple groups, maximal normal subgroups, the center of a group, the commutator subgroup of a group, why the converse of Lagrange's Theorem is false.
3, 5, 19, 20, 34, 37, 40

Section 26. Homomorphisms and factor rings. Ring homomorphisms (again) and what they preserve, kernels of such, ideals, quotient rings, the "fundamental homomorphism theorem" for rings.
3 (note: each ideal must be an additive subgroup), 4, 10 (TF TF TF TT TT), 15, 18 (show 1 is in the kernel), 20 (use the binomial theorem), 26 (this is called the annihilator of a - is that cool or what?), 27 (usual proof), 37

Section 27. Prime and maximal ideals. Maximal ideals M and the quotient rings R/M (which are fields), prime ideals P and the quotient rings R/P (which are integral domains), why each ring with 1 contains a subring isomorphic to either the integers or the integers modulo n, how each field contains a subfield isomorphic to either the rational or the integers modulo p, applications to the polynomial ring R=F[x] where F is a field.
2 (see Thms 19.11, 19.12, 27.9), 5, 14 (FT TF TT TF TF), 15, 18, 19 (Eisenstein), 24 (see Thm 19.11), 30 (use the facts that all ideals are principle and that all elements of the ideal generated by f(x) have degree at least as great as the degree of f(x) to show f(x) is irreducible), 37

Section 29. Introduction to extension fields. Extension fields, Kronecker's Theorem, algebraic and transcendental elements over a field (special case: algebraic and transcendental numbers), monic polynomials, irreducible (monic) polynomial for an element over a field, simple extensions.
4, 6, 16 (hint: (x²)³=(x³)²), 23 (TT TT FT FT FT), 24 (for (a), see problem 16), 26 (this is much easier than it sounds), 30 (Thm 29.18), 33 (pg 270 Case II)

Section 30. Vector spaces. Vector spaces, linear independence, span, bases, dimension, and what all this has to do with extension fields.
1 - 9, 15 (TF TT FF FT TT), 16 (see last paragraph on page 173), 21 (so I guess the uniqueness stuff is important), 23 (so all n-dimensional vector spaces over the same field are essentially the same)

Section 31. Algebraic extensions. Algebraic extensions, why finite extensions are algebraic, what this formula means: [K:F] = [K:E][E:F], algebraically closed field, Fundamental Theorem of Algebra, partially ordered sets (posets), Zorn's Lemma and a proof that every field has an algebraic closure.
1, 2 (Ex. 31.9), 3, 4, 6 (Ex. 31.9), 7, 11 (Ex. 31.9), 19 (FT FT FT FF FF), 23, 24, 29, 30 (Note: these last four problems use Thm. 31.4)

Section 32. Geometric constructions. What abstract algebra has to do with straightedge and compass constructions in the plane, and three classic impossible constructions.

Here are some links with instructions on how to do the "basic constructions" Fraleigh mentions in this section. Both come from the Geometry Construction Reference Page in Whistler Alley Mathematics .

• Given point P on line k, construct a line through P, perpendicular to k.
• Given a line and a point, construct a line through the point, parallel to the given line.

Section 33. Finite fields. nth roots of unity, primitive nth roots of unity, why every finite field must have prime-power order, why there is a finite field for every prime-power order, and why all such fields of the same order are isomorphic (the Galois field of order pⁿ, GF(pⁿ)).
1, 4, 7, 8 (TF TF TF TT FT), 9

Section 38. Free abelian groups. Groups with bases (known as free abelian groups), why each is isomorphic to a direct product of a bunch of copies of the integers, and a fairly technical proof of the Fundamental Theorem of Finitely Generated Abelian Groups (don't get too hung up on this part of the section).
2, 3, 7, 8 (TT TT TF FT TF), 10 (assume a nonzero element a = n₁x₁ + n₂x₂ + ... + n_kx_k has order m > 0 and get a contradiction), 14 (use good old Theorem 5.14; you get the identity for free)

Section 39. Free groups. Reduced words, free group F[A] generated by a set A, rank, homorphisms of free groups, the fact that every group is a homomorphic image of a free group.
1, 2, 3 - 6 (note: for "onto," you must hit a generating set in the codomain; also, recall that the homomorphic image of an abelian group is abelian), 10 (TF FT F? FT FT ... I think (f) is supposed to say, "No free group is free abelian," which is false (Z is both)), 11 (for (d), use FTFGAG)

Section 40. Group presentations. Presenting groups by generators and relations. (Note: Since we didn't cover sections 36 or 37, the references to Sylow theory in Examples 40.5 and 40.6 won't make any sense to you, so don't bother with those examples.)
1, 3, 8 (TT FF FT TF TF), 10 (Don't use Exercise 13; rather, just play around with the presentation and see what you can see. For example, since ba = a²b, you can write every element of this group as aⁿb^m; use the other relations to figure out what n and m can be.)

Section 12. Plane isometries. The full group of isometries of the plane, the four different types of isometries of the plane (translations, rotations, reflections, glide refelections), subgroups of the full group of isometries of the plane that carry a subset of the plane onto itself, finite subgroups of the group of isometries of the plane (either cyclic or dihedral), symmetry types of discrete frieze patterns, and symmetry types of wallpaper patterns.
2, 4 - 10, 14, 16 - 18, 21, 24 - 30

Section 16. Group action on a set. Group action on a set, faithful actions, transitive actions, isotropy subgroups (G_x is also called the stabilizer of x), orbits of elements.
1 - 3, 7, 8 (FT FT FT TF TT), 11 - 13

Section 45. Unique factorization domains. Divisibility, associates, irreducibles, primes, gcds, primitive polynomials, unique factorization domains (UFDs), principle ideal domains (PIDs), the Ascending Chain Condition (which holds in a PID). [Note: Something for which the ACC holds is also known as a Noetherian, after Emmy Noether, while something for which the DCC holds is also known as Artinian, after Emil Artin. Just a piece of history for you.] The proofs that every PID is a UFD, and if D is a UFD, then so is D[x]. Here is an informative page on UFDs, PIDs, and Euclidean domains that might be worth a look.
3, 5, 7, 9, 14, 16, 21 (TT TF TF FT FT), 25, 27, 28

Section 46. Euclidean domains. Euclidean norms (also known as Euclidean valuations), Euclidean domains (integral domains where a generalization of the division algorithm holds), the Euclidean algorithm (used for finding gcds), why every Euclidean domain is a PID (and hence a UFD).
1, 2, 6, 12, 13 (TF TF TT TF TT), 15, 18 (make v(a)=1 for all nonzero a), 22