MTH 610(sec 201) Galois Theory Syllabus Spring 2012

CRN 4307

Prerequisites: MTH 450 and MTH 452 , also must have taken MTH 300

Course Objectives : To learn about Galois groups on extension fields, splitting fields and

solvability of polynomial equations.

Meeting time : T , R 12:30-1:45pm Room 518 Smith Hall

Instructor : Dr. Alan Horwitz Office : Room 741 Smith Hall

Phone : (304)696-3046 Email :

Text : Galois Theory, 3rd edition, Ian Stewart, Chapman and Hall/CRC Press

Grading : homework (and possibly labs) 35% (175points)

2 major exams 40% (200 points)

(if we have a 3rd exam, then I'll count the highest two)

final( comprehensive ) exam 25% (125 points)

total : 500 points

Final exam date : Thursday May 1, 2012 at 12:45-2:45 pm

General Policies :

Attendance is required . You are responsible for reading the text, working the exercises, coming to office hours for help when you’re stuck, and being aware of the dates for homework assignments and major exams as they are announced. Late homework will be penalized.

Exam dates will be announced at least one week in advance. Makeup exams will be given only if you have an acceptable written excuse with evidence and/or you have obtained my prior permission.

Make ups are likely to be more difficult than the original exam and must be taken within one calendar week of the original exam date. You can’t make up a makeup exam: if you miss your appointment for a makeup exam, then you’ll get a score of 0 on the exam.

If you anticipate being absent from an exam due to a prior commitment, let me know in advance so we can schedule a makeup. If you cannot take an exam due to sudden circumstances, you must call my office and leave a message on or before the day of the exam!

In borderline cases, your final grade can be influenced by factors such as your record of attendance, whether or not

your exam scores have been improving during the semester, and your class participation.

Attendance:

Regular attendance is expected ! If your grade is borderline, then good attendance can result in attaining a higher grade. Likewise, poor attendance can result in a lower grade.

Having more than 3 weeks worth of unexcused absences (i.e., 6 lectures ) will automatically result in a course grade of F! Being habitually late to class will count as an unexcused absence for each occurrence. Carrying on conversations with your neighbor could be counted as being absent. Walking out in the middle of lecture is rude and

a distraction to the class ; each occurrence will count as an unexcused absence. If you must leave class early for

a doctor’s appointment , etc., let me know at the beginning and I’ll usually be happy to give permission.

Absences which can be excused include illness, emergencies, or official participation in another university activity.

Documentation from an outside source ( eg. coach, doctor, court clerk…) must be provided. If you lack documentation, then I can choose whether or not to excuse your absence.

Sleeping in Class :

Habitual sleeping during lectures can be considered as an unexcused absence for each occurrence. If you are that tired, go home and take a real nap! You might want to change your sleeping schedule, so that you can be awake for class.

Cell Phone and Pager Policy :

Unless you are a secret service agent, fireman, paramedic on call or a drug dealer, all electronic communication devices such as pagers and cell phones should be shut off during class. Alternatively, you can use the heel of your boot to grind your cell phone into the linoleum. Violation of this policy can result in confiscation of your device and your forced participation in a study on the deleterious health effects of excessive cell phone use.

Week

/

Dates

Spring
2012 / Sections covered and topics
1 / 1/9-
1/13

2

/ 1/17-
1/20 /
3 / 1/23-
1/27
4 / 1/30-
2/3
5 / 2/6-
2/10
Week /

Dates

Spring
2012 / Sections covered and topics
7 / 2/20-
2/24

8

/ 2/27-
3/2 /
9 / 3/5-
3/9

10

/ 3/12-
3/16
(last day
to drop
on 3/16) /
11 / 3/26-
3/30
12 / 4/2-4/6
(Assessment day on 4/6)
Week /

Dates

Spring
2012 / Sections covered and topics
13 / 4/9-
4/13
14 / 4/16-4/20

15

/ 4/23-
4/27 /

MTH 610(sec 201) Topics in Galois Theory(Ian Stewart)

1/10/12

1.1 complex numbers

1.2 review of groups, rings, fields, subrings and subfields,

homomorphisms and isomorphisms

monomorphisms vs. automorphisms

nth-roots of unity

1.3 solving equations as a motivation for new number systems

1.4 solving quadratic equations by completing the square

using Tschirnhaus transformations to rewrite quadratic, cubic, quartic, quintic

equations

Cardano’s Formula for solving cubic equations and the strange form of it solutions

solving quartic equations

why Tschirnhaus transformation method does not work for solving quintic equations

2.1 properties of polynomials over complex numbers

polynomials in several variables

notation for degree of a polynomial in one variable

2.2 Fundamental Theorem of Algebra

winding number of a loop in the plane stays constant under slight deformations

proving the Fundamental Theorem of Algebra

2.3 Remainder Theorem

completely factoring any complex polynomial over the complex numbers

3.1 Division Algorithm

greatest common factor of two polynomials

using the Euclidean Algorithm to find a greatest common factor

expressing greatest common factor as a linear combination of the two polynomials

3.2 reducible vs. irreducible polynomials over a subring

polynomials can be factored as a product of irreducible polynomials

coprime polynomials have greatest common factor of 1

uniqueness of factorizations into product of irreducible polynomials

3.3 Gauss Lemma on irreducibility over integers and rational numbers

3.4 Eisenstein’s Criterion for irreducibility over the rationals

3.5 Congruence modulo n

Units in integers modulo n

Euler phi function

Using irreducibility in modulo n to show irreducibility over the integers

3.6 zeros of multiplicity m for polynomials

the number of distinct zeros is no larger than the degree

4.1 field extension monomorphism

subfield generated by a subset of a field

subfield generated by adjoining a subset of a field extension

4.2 rational expressions in one and in n variables

4.3 simple field extensions

commutative diagram to represent an isomorphism between two field extensions

5.1 algebraic over a field vs. transcendental over a field

5.2 monic polynomials

minimal polynomial of an algebraic element in a field extension

any irreducible monic polynomial is a minimal polynomial for some complex number

5.3 reduced form of a polynomial modulo m

ideals generated by irreducible polynomials give quotient rings which are fields

MTH 610(sec 201) Topics in Galois Theory(Ian Stewart)

1/10/12______

5.4 simple algebraic extensions are isomorphic to quotient rings

generated by a minimal polynomial

algebraic extensions with same minimal polynomial are isomorphic

degree of a simple algebraic extension matches degree of the minimal polynomial

extending an isomorphism between fields to an isomorphism between simple algebraic

extensions of the fields

6.1 viewing a field extension as a vector space over the field

the degree of a field extension over a field

6.2 the short and regular Tower Laws for degrees of field extensions over subfields

degree of a simple transcendental extension over its field is infinite

degree of a simple algebraic extension over a field is equal to the

degree of the minimal polynomial

finite extensions are algebraic extensions formed by attaching

finitely many algebraic elements

7.1 an angle can be trisected with a compass and a marked ruler

definition of a point being constructible using a compass and unmarked ruler from an

original point

tower of subfields created by adjoining coordinates of constructible points

coordinates of constructible points are zeros of quadratic polynomials

fields created by adjoining constructible points have degrees which are powers of 2 over

the field generated by the coordinates of the original point

7.2 Wantzel’s Theorem on not being able to “duplicate” a cube with

ruler and compass constructions

Wantzel’s Theorem on not being able to trisect a 60 degree angle

with ruler and compass constructions

being unable to square a circle with ruler and compass constructions

8.2 permutations of the zeros preserve the coefficients of the polynomial equation

and form a subgroup of a symmetric group

the Galois group of zeros of a polynomial equation

8.3 describing subgroups of a Galois group by the fields which they leave fixed

subgroups of Galois group determine whether the polynomial equation

can be solved by radicals

8.4 K-automorphisms of a field extension L

Galois correspondence between subgroups of a Galois group and subfields of the field

extension containing all zeros(which correspond to the Galois group)

8.5 all K-automorphisms of a field extension are defined as the Galois group

8.6 fixed fields of subgroups of the Galois group

Galois correspondence between subgroups of a Galois group and their fixed fields

8.7 writing coefficients of a polynomial as elementary symmetric polynomials of the zeros

tower of subfields needed for solubility by Ruffini radicals

refining a tower of subfields to make degrees of consecutive field extensions prime

lemmas on cyclic quotient groups of symmetric group and alternating group

general polynomials of degree 5 or higher are insoluble by Ruffini radicals

Klein 4-group

the discriminant of a polynomial

8.8 solvability by radicals implies solubility by Ruffini radicals

general polynomial equations of degree 5 or higher are not solvable by radicals

theorem of natural irrationalities

the height of a radical extension over a field is always finite

factorability of the minimal polynomial of an element in an extension field

MTH 610(sec 201) Topics in Galois Theory(Ian Stewart)

1/10/12______

9.1 splitting field of a polynomial

isomorphisms between fields extend to isomorphisms between splitting fields

9.2 ……………..to be continued………………..