Concrete Algebra With a View Toward Abstract Algebra

1 The integers
2 Mathematical induction
3 Greatest common divisors
4 Prime numbers
5 Modular arithmetic
6 The Chinese remainder theorem
7 Secret messages
8 Rational, real and complex numbers
9 Polynomials
10 Real polynomials, complex polynomials
11 Factoring polynomials
12 Fields
13 Field extensions
14 Formal power series
15 Resultants and discriminants
16 Permuting roots
17 Morphisms
18 Galois theory
19 Plane algebraic curves
20 Where plane algebraic curves intersect
21 Quotient rings
22 Field extensions and algebraic curves
23 The projective plane
24 Algebraic curves in the projective plane
25 Counting intersections of curves
26 Families of plane algebraic curves
27 Elliptic curves
28 The tangent line
29 Inflection points
30 Conics and quadratic forms
31 Projective duality
32 Polynomial equations have solutions
33 Blow up
34 Schubert calculus
35 More projective planes
Hints
Bibliography
List of notation
Index