Algebra I (Math 520A)
Topics and Assignments
Spring Semester, 2007
End of Semester Schedule
- Tue., May. 15:
- Final Examination: 3:30 - 5:30 p.m.
- Mon., May. 14:
- Office Hours: 2:30 - 3:30 p.m.
- Fri., May. 11:
- Office Hours: 2:30 - 3:30 p.m.
PDF and DVI
(requires TeX software) versions
of this page are available for printing.
For each date there is a list of topics covered (or to be covered).
For each meeting your assignment is to be ready for a quiz on recently
covered definitions and statements of theorems.
Each written assignment will be announced one week before it is due.
- Mon., May. 7:
- Last meeting prior to the final exam.
Written Assignment No. 5 (also available as PDF) is due.
- Fri., May. 4:
- The notion of normal extension. A finite extension is Galois if and
only if separable and normal. In a Galois extension there is a
bijective correspondence between intermediate fields and subgroups of
the Galois group. A finite extension L of degree n over K is
Galois if and only if there is an irreducible polynomial f of degree
n with no multiple root in “the” algebraic closure of K such
that L is “the” splitting field of f. In any finite separable
extension there are only finitely many intermediate fields.
- Wed., May. 2:
- Completed the proof that every finite separable extension is
simple. If L/K is a Galois extension with group G, then
for every subgroup H of G with E the subfield of L fixed
by H, the extension L/E is Galois with group H. Two
consequences of that: (1) A finite extension L / K is Galois if
and only if, with G = Aut(L/K), one has K = L^{G} (the
case H = G). (2) Every (finite) Galois extension is separable.
- Mon., Apr. 30:
- The notion of element separably algebraic over a field, and separably
algebraic extensions. Examples. A finite extension L/K is separable
if and only if the number of K-algebra morphisms of L in “the”
algebraic closure of K is equal to the extension degree of L over
K. Every finite separable extension is simple.
- Fri., Apr. 27:
- Example: The splitting field E over Q of the polynomial
t^{3} - 2 is Q(SQRT[3]{2}, rho) where rho is a (complex)
root of t^{2} - t + 1. E/Q is a Galois extension with
Aut_{Q}(E) ~= S_{3}. Theorem: Every finite extension
of a finite field with q elements is a Galois extension with
cyclic Galois group generated by alpha :--> alpha^{q}.
- Wed., Apr. 25:
- A finite sequence of distinct homomorphisms from a group G to the
multiplicative group F^{*} of a field F is linearly independent when
regarded as a finite sequence of elements of the vector space over F
of arbitrary maps from G to F (with vector space operations
defined pointwise). If G is a subgroup of the group
Aut_{K}(L) for a finite field extension L / K, then
A finite extension L / K is called a Galois extension if
- Mon., Apr. 23:
- Every finite subgroup of the multiplicative group of a field is
cyclic. The number of elements in a finite field must be a prime
power. For every prime power q there is one and (up to non-unique
isomorphism) only one finite field having q elements. The unique
field with q = p^{r} elements (p prime, r >= 1) may be characterized
as “the” splitting field of the polynomial t^{q} - t \in
F_{p} [t].
Written Assignment No. 4 (also available as PDF) is due.
- Fri., Apr. 20:
- Notion of splitting field of a polynomial with coefficients in a
field. Any field in which a polynomial splits is (in a non-unique
way) an extension of any splitting field. Existence and uniqueness
(up to non-unique isomorphism) of a splitting field for a given
polynomial. Definition of “the” algebraic closure of a field.
Every field has an algebraic closure that is unique up to non-unique
isomorphism.
- Wed., Apr. 18:
- Notion of algebraic field extension. The long extension in a finite
tower of algebraic extensions is an algebraic extension. An extension
L of K is finite if and only if there exist finitely many elements
alpha_{1}, … alpha_{m} algebraic over K such that
L = K(alpha_{1}, …, alpha_{m}). In any extension L of K the
set of all elements in L that are algebraic over K is a subfield
of L that is called the algebraic closure of K in L.
Definition and equivalent criteria for an algebraically closed field.
Existence of an algebraically closed extension of any field.
- Mon., Apr. 16:
- Prime and maximal ideals in a commutative ring.
An ideal I in a commutative ring A is prime if and only if A/I
is a domain; I is maximal if and only if A/I is a field. For a
field extension L / K and an element alpha \in L let
varphi : K[t] --> L denote the ring morphism given by the substitution
of alpha for t. Then
|
K[alpha] = Imvarphi ~= K[t]/Kervarphi .
|
Because K[alpha] is finite-dimensional over K if and only if
Kervarphi ≠ (0), alpha is transcendental over K if and
only if K[alpha] ~= K[t], and alpha is algebraic over K if
and only if K[alpha] = K(alpha), i.e., if K[alpha] is a field.
When alpha is algebraic over K there is a unique irreducible monic
polynomial f \in K[t] such that f(alpha) = 0.
- Fri., Apr. 13:
- Extension degrees multiply in a tower. Elements of a field algebraic
or transcendental over a subfield. For a field extension L / K
an element alpha \in L is algebraic over K if and only if
K(alpha) is a finite extension of K. Examples.
- Wed., Apr. 11:
- Comments on assignment no. 3. Ring and field adjunction inside a
field. Finite extensions and extensions of finite type.
F[t]/f(t)F[t] is a finite extension of F when f is
a polynomial in F[t] irreducible over F.
- Mon - Mon., Apr. 2 - 9:
- University Recess: no meetings
- Fri., Mar. 30:
- The characteristic and characteristic subring of a ring.
The p^{th}-power endomorphism of a commutative ring in
prime characteristic p; this endomorphism is an automorphism
in the case of a finite field of characteristic p.
Written Assignment No. 3 (also available as PDF) is due.
- Wed., Mar. 28:
- Overview of field extensions. The group of relative automorphisms.
Examples.
- Mon., Mar. 26:
- Normal series and composition series. The Jordan-Holder theorem.
Examples.
- Fri., Mar. 23:
- The notion of invariant or characteristic subgroup.
Examples: The center and the commutator (or derived)
subgroup of a group. G^{ab} ~= G/Der(G) is universal
for morphisms from G to an abelian group.
- Wed., Mar. 21:
- Midterm Test.
- Mon., Mar. 19:
- Questions prior to the test.
The concept of simple group.
- Fri., Mar. 16:
- The quaternion group as an example of a group given by generators
and relations.
Complete classification of groups of orders up to 8.
- Wed., Mar. 14:
- A finite abelian group is the direct product of its Sylow subgroups.
Thus the classification of finite abelian groups reduces to the
classification of finite abelian p-groups. There is a
course supplement (also available as PDF) following this approach
that is accessible at this point.
The classification of finite abelian groups will be revisited in
Math 520B as part of the study of finitely-generated modules over principal
ideal domains.
- Mon., Mar. 12:
- Proof of the Sylow theorems.
Written Assignment No. 2 (also available as PDF) is due.
- Fri., Mar. 9:
- Application of the Sylow theorems to the classification of groups of
order 105
- Wed., Mar. 7:
- The concept of a p-group. Application of Cauchy's theorem: a finite
p-group is the same thing as a group of order p^{r}. Kernel of the
action of a group by translation on a quotient set; normality of
a subgroup having smallest prime index. Classification of groups of
order pq where p and q are distinct primes. Announcement of the
Sylow theorems.
- Mon., Mar. 5:
- Issue: the existence of subgroups in a finite group. The case of
cyclic groups. If X is a finite G-set and |G| a power of
p, a prime, then the number of fixed points is congruent to
|X| modulo p. Cauchy's theorem on the existence of
elements of prime order.
- Fri., Mar. 2:
- The second and third isomorphism theorems as applications of the
universality of the quotient. Examples.
- Wed., Feb. 28:
- The class formula and its application to p-groups. The center of a
group cannot have prime index. Every group of order p^{2} is abelian.
The subgroup of a group generated by a subset. In a group G if H
is a subgroup and N a normal subgroup, then
= NH,
and NH is semi-direct if and only if N cap H =
. Example:
SL_{2}(Z) is generated by two elements with orders 3 and
4.
- Mon., Feb. 26:
- Splittings. Example: Split and non-split homomorphisms on the
Heisenberg group of a vector space. Cyclic and dihedral groups.
- Week Feb 19 - 23
- University Recess: no classes.
- Fri., Feb. 16:
- Internal and external semi-direct products as equivalent notions.
Written Assignment No. 1 (also available as PDF) is due.
- Wed., Feb. 14:
- No Meeting.
The University has announced that, due to severe snow conditions,
all day and evening classes on February 14 are cancelled. As a result,
the due date for Written Assignment No. 1 has been postponed to Friday,
Feb. 16.
- Mon., Feb. 12:
- Transitive actions and factor sets. Actions by
automorphisms.
- Fri., Feb. 9:
- The notion of group action. Examples. Actions lurking in the
group of rotations of the cube.
- Wed., Feb. 7:
- Universality of the quotient for rings. Examples. The notion of
algebra over a ring.
- Mon., Feb. 5:
- Kernels and images, of morphisms for rings and modules over a ring.
Ideals in a ring, and the quotient of a ring by a two-sided ideal.
- Fri., Feb. 2:
- Universality of the quotient morphism and the first isomorphism
theorem.
- Wed., Jan. 31:
- Kernels and images of morphisms for groups, normal subgroups,
and the quotient construction.
- Mon., Jan. 29:
- Hamilton's quaternions as an example of a division ring that is
not a field. Left, right and two-sided modules over a ring and
morphisms of modules.
- Fri., Jan. 26:
- Rings, zero divisors, division rings, fields, and ring morphisms.
- Wed., Jan. 24:
- Left and right translations, invertibility in a monoid, groups.
- Mon., Jan. 22:
- Two different understandings of the notion of sub-object. A set
with binary operation as a sub-object of another.
Subsemigroups, submonoids, the monoid of endomorphisms of a semigroup
as a submonoid of the monoid of self-maps of the underlying set.
- Fri., Jan. 19:
- Binary operations, associativity, commutativity, semigroups, monoids,
and the corresponding morphisms. Examples: Addition and
multiplication of n \times n matrices, the Lie bracket, and the
monoid of endomorphisms of a set under composition. For n \times n
real matrices x, y the function
defines a symmetric bilinear form, not a commutative binary operation.
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