| chapter ⅰ set theory and number theory . 1 set theory 2 unique factorization theorem 3 congruence 4 chinese remainder theorem 5 complex integers 6 real numbers and p-aclic numbers chapter ⅱ group theory 1 definitions 2 the transformation groups on sets 3 subgroups 4 normal subgroups and inner automorphisms 5 automorphism groups 6 p-groups and sylow theorems 7 jordan-holder theorem 8 symmetric group sn chapter ⅲ polynomials .. 1 fields and rings 2 polynomial rings and quotient fields 3 unique factorization theorem for polynomials .4 symmetric polynomial, resultant and discriminant 5 ideals chapter ⅳ linear algebra 1 vector spaces 2 basis and dimension 3 linear transformation and matrix 4 module and module over p.i.d 5 jordan canonical form 6 characteristic polynomial 7 inner product and bilinear form 8 spectral theory chapter ⅴ polynomials in one variable and field theory 1 algebraically closed field 2 algebraic extension 3 algebraic closure 4 characteristic and finite field 5 separable algebraic extension 6 galois theory 7 solve equation by radicals 8 field polynomial and field discriminant 9 luroth's theorem appendix a1 set theoretical notations a2 peano's axioms a3 homological algebra index ... |
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