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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 : [email protected]
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
1
Dates
Spring
2012
1/91/13
2
1/171/20
3
1/231/27
4
1/302/3
5
2/62/10
Sections covered and topics
Week
7
Dates
Spring
2012
2/202/24
8
2/273/2
9
3/53/9
10
3/123/16
(last day
to drop
on 3/16)
11
3/263/30
Sections covered and topics
12
4/2-4/6
(Assessment
day on 4/6)
Week
Dates
Spring
2012
13
4/94/13
14
4/16-4/20
15
4/234/27
Sections covered and topics
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
4.3
5.1
5.2
5.3
rational expressions in one and in n variables
simple field extensions
commutative diagram to represent an isomorphism between two field extensions
algebraic over a field vs. transcendental over a field
monic polynomials
minimal polynomial of an algebraic element in a field extension
any irreducible monic polynomial is a minimal polynomial for some complex number
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………………..