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{\bf\centerline {Math 362 Syllabus}}
{\bf \centerline{Spring, 2001}}
{\centerline{Instructor: Warren Wong}}
\vskip .15cm
{\bf Textbook} Galois theory by Ian Stewart, 2nd edn, Chapman and Hall,
1998.\vskip .15cm {\bf Syllabus}
\vskip .15cm {\bf 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 Ideals in polynomial rings over fields
\vskip .15cm {\bf 2 Factorization of polynomials (review)} 2.1
Irreducibility 2.2 Tests for Irreducibility (Symmetric polynomials were
considered later in the course, with the general polynomial)
\vskip .15cm {\bf
3 Field Extensions} 3.1 Field extensions 3.2 Simple extensions 3.3 Constructing
simple extensions 3.4 Classifying simple extensions
\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 (trisection of angle of an
equilateral triangle, regular septagon) \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
monomorphisms\vskip .15cm {\bf 10 Monomorphisms, automorphisms and normal
closure} 10.1 $K$-monomorphisms 10.2 Normal closures \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)
\vskip .15cm {\bf 13 Soluble and simple groups}
13.1 Soluble Groups (definition, and proof that $S_n$ is solvable if and
only if
$n\leq 4$)
\vskip .15cm {\bf 14 Solution of equations by radicals} 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 \vskip .15cm {\bf 15 The
general polynomial} 15.2 The general polynomial (Non-solvability 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)
\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) \vskip .15cm {\bf 19 The
fundamental theorem of algebra} Proved the fundamental theorem of algebra using
ideas from Galois theory
\vskip
.15cm
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