MATH 535: Algebra
| Lecture |
Topics Covered |
Reading Assignment |
Homework |
Homework Solutions |
|---|
| Week 1 | Definition of a group and examples, including Sn and symmetries
of geometrical objects including Dn, cube and dodecehedron.
Classification of Platonic solids, permutation groups, symmetry
group of regular tetrahedron. Examples of subgroups,
Lagrange's
theorem and applications to orders. |
2.2 - 2.4 |
HW #1 due 9/13/04 |
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| Week 2 |
Definition of homomorphism, etc., normal subgroups, center,
inner automorphisms.
Quotient groups and First isomorphism theorem, examples.
Isomorphism theorems, G/Z = Inn (G) , Zmn = Zm X Zn for
(m, n) = 1. Subgroups and automorphisms of cyclic
groups, canonical homomorphisms and correspondences between subgroups.
|
2.5, 2.6
|
HW #2 due 9/20/04
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| Week 3 |
Groups actions, examples, orbits and stabilizer sugroups.
Cayley's theorem.
Conjugacy classes, class equation, p-groups. |
2.7 |
HW #3 due 9/27/04
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| Week 4 |
Cauchy's theorem, A5 is a simple group, towards Frobenius-Burnside
theorem. Frobenius-Burnside theorem and applications.
|
2.7 |
HW #4 due 10/04/04
|
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| Week 5 | Commutative rings, basics, field of fractions, polynomials.
Division algorithm, unique factorization for F[X].
Irreducible polynomials. Gauss lemma. Eisenstein irreducibility criterion. Cyclotomic polynomials.
Homomorphisms, ideals, PID, Euclidean rings. |
3.2 - 3.6 |
HW #5 due 10/11/04
|
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| Week 6 | Linear algebra. Quotient rings,
Isomorphism theorems for rings.
|
3.7, 3.8 |
HW #6 due 10/18/04 |
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| Week 7 | F[X]/p(X) field iff p(X) irreducible.
Finite fields. Prime fields, characteristics, algebraic and
transcentental extensions, degrees of extensions. |
3.8 |
|
|
| 10/15/04 |
No Class, Study Day |
3.8 |
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| Week 8 | Minimal polynomial. F(α) ≈ F[X]/p(X) where p(X) is
minimal polynomial of α. Splitting fields, dimension. Examples, finite fields.
Uniqueness of splitting fields up to isomorphism. |
3.8 |
|
|
| 10/20/04 | Midterm Exam. |
5-7pm, |
118 Sackett |
|
| Week 9 |
Definition of Galois group. Separable polynomials, separable extensions.
Inseparable extension.
Some computations of Galois groups.
|
4.1 |
HW #7 due 11/01/04 |
|
| Week 10 | Frobenius automorphism. Galois group of a finite field
is cyclic.
Normal extensions.
|
4.1 |
HW #8 due 11/08/04 |
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| Week 11 |
Characterization of normal extensions. Galois extensions.
Theorem of the primitive element.
Fixed fields and Galois groups. The fundamental theorem of Galois theory.
|
4.2 |
HW #9 due 11/17/04 |
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| Week 12 | Cyclotomic fields, cyclotomic polynomials are irreducible.
If ζ is a primitive n-th root of unity then Gal(Q(ζ)/Q)
is isomorphic to U(Zn) , examples n = 8, 20. |
3.4 |
HW #10 due 11/29/04
|
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| Week 13 | Cyclic and Kummer extensions. Characters. Dedekind's lemma
on linear independence of distinct characters. |
4.2 |
HW #11 due 11/06/04 |
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| Week 14 | Solvability by radicals. Radical extensions.
Ruler-compass constructions,
Solvable groups.
solvable by radicals iff Galois group solvable.
Some generating sets in Sn. Sufficient condition on the roots for Galois group
S5.
Explicit f(X) ∈ Q[X] with Galois group S5.
|
4.2 |
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| Week 15 |
The discriminant. The Galois group of a cubic over Q. The Fundamental Theorem of Algebra.
|
4.2 |
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