Integers mod n. Finite fields GF(p). Matrix algebra. Vector Spaces. Linear codes. Linear transformations. Invariant subspaces. Groups. Permutation groups (groups acting on structures). Determinants. Rings . Structure of finite commutative rings. Rings of polynomials and rings of power series, Groebner bases, symmetric functions. Resultants, factorizing polynomials. Rings of fractions, partial fractions; approximating power series by rational functions. Extension fields, basic structure of finite fields; discrete log, theorem of algebra. Automorphisms of GF(q); trace; normal bases, optimal normal bases and fast hardware.. Cyclic codes, BCH codes, Golay code, Reed-Solomon codes. p-adic numbers, lifting factorizations mod p to factorizations mod pn. Discrete Fourier transform, applications to cylic codes, Kronecker product and fast transforms. Linear systems and control theory.
