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{\bf\centerline {Math 362 Syllabus}}
{\bf \centerline{Spring, 1999}}
{\centerline{Instructor; M. Dyer}}
\vskip .15cm
{\bf Textbook} Galois theory by Ian Stewart, 2nd edn, Chapman and Hall, 1992.
This text was out of print but I managed to locate a copy for the  one
student in the class. {\vskip .15cm\bf Remark}  I
ran the course as a reading course (there was only one student), and spent class
time answering questions, giving complements to the reading, doing examples and
exercises, and having the student do through exercises. Additional homework
exercises were assigned as well. In this way, most of the unstarred exercises
(and a few starred ones) in Chapters 1--5 and 7--16 in the text were covered.
\vskip .15cm {\bf Syllabus}

\vskip .15cm {1 Ring theory (review)} 1.1 General properties of
rings 1.2 Characteristic of a field  1.3 Fields of fractions 1.4 Polynomials
1.5 Euclidean algorithm
     \vskip .15cm {\bf 2 Factorization of polynomials (review)} 2.1
Irreducibility  2.2 Test for Irreducibility (Gauss Lemma,
reduction mod p, Eisenstein's criterion) 2.3 Zeros of polynomials

 \vskip .15cm {\bf
3 Field Extensions} 3.1 Field extensions 3.2 Simple extensions 3.3 Constructing
simple extensions 3.4 Classifying simple extensions The Euclidean algorithm 3.2
Roots of polynomials 3.3 Polynomials with integer coefficients 
\vskip .15cm {\bf 4
The degree of an extension} 4.1 The tower law (multiplicativity of extension
degrees) 4.2 Algebraic Numbers \vskip .15cm {\bf 5 Ruler and Compass} 5.1
Algebraic formulation 5.2 Impossibility proofs (duplication of cube,
trisection of angle of an equilateral triangle, squaring the circle (last
given modulo the proof of transcendence of $\pi$ in Chapter 6),
 stated but did not prove Gauss' theorem on constructability of
regular $n$-gon (Chapter 17 of book)  \vskip .15cm {\bf 7 The idea behind Galois
theory} \vskip .15cm{\bf 8 Normality and separability} 8.1 Splitting fields 8.2
Normality 8.3 Separability 8.4 Formal Differentiation  \vskip .15cm {\bf 9} Field
degrees and group orders 9.1 Linear independence of homomorphisms (and
application to proof of Artin's theorem that degree of a field over fixed
subfield of a finite group is the group order)\vskip .15cm {\bf 10 Monomorphisms,
automorphisms and normal closure} 10.1 $K$-monomorphisms  (results on number of
monomorphisms of a finite  field extension into its normal closure)\vskip .15cm
{\bf 11 The Galois correspondence} 11.1 The fundamental theorem (proof) \vskip
.15cm {\bf 12.1 A specific example} (detailed examination of fundamental theorem
for splitting field of $t^4-2$ over the rationals) Discussed in class in detail
a second ``large''  example ($t^5-2$ over the rationals).
\vskip .15cm {\bf 13 Soluble and simple groups (review)}
13.1 Soluble Groups 13.2 Simple groups 13.3 $p$-groups, 
simplicity of alternating group $A_n$ for $n\geq 5$)\vskip .15cm {\bf 14
Solution of equations by radicals} To motivate the problem, I described series
of substitutions leading to solutions of cubics and quartics over fields of
characteristic not two or three  (this is discussed using Galois theory in 15.3
of text, but  I took a more naive approach) 14.2 Radical extensions; proved
the  theorem that polynomial equations in characteristic zero are solvable by
radicals only if their Galois group is solvable  14.3
An insoluble quintic ($t^5-6t+3$ over the rationals)\vskip .15cm {\bf 15 The
general polynomial}15.1 Transcendence degree 15.2 The general polynomial Showed
insolubility by radicals of general polynomial of degree $n\geq 5$ 
15.3 Solving quartic equations (Showed solvable Galois group implies  an
equation solvable by radicals in characteristic zero; did not discuss
explicit solution of cubics and quartics using ideas from Galois theory, after
giving the earlier naive treatment of them)  
\vskip .15cm {\bf 16  Finite fields}16.1 Structure of finite fields
16.2 Multiplicative group (also examined in detail the Galois
correspondence for extensions of finite fields (left as an exercise in
the text) and obtained the formula for the number of irreducible
polynomials of each degree over a finite field)\vskip .15cm {\bf 19  The
fundamental theorem of algebra} 19.1 Ordered fields and their extensions
(proved the fundamental theorem of algebra using ideas from Galois theory).
  \vskip
.15cm



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