Group Theory Review Group actions Sylows theorems
Rings and Modules Review ED PID UFD prime and maximal ideals and modules Structure theorem of finitely generated modules over PID Tensor product of modules
Linear Algebra Review Vector spaces direct sums tensor products Linear transformations and Matrices Determinants Diagonalizability and Nilpotence Rational canonical form Jordan form Bilinear forms Inner product spaces unitary self-adjoint normal and isometric transformations Spectral theorem
Galois Theory Algebraic and transcendental extensions algebraic closure splitting fields and normal extensions separable inseparable and purely inseparable extensions finite fields Galois extensions and Galois groups Fundamental theorem of Galois theory cyclic extensions solvability by radicals constructibility of regular n-gons cyclotomic extensions
Representation Theory Introduction to multilinear algebra Review of linear algebra multilinear forms tensor products wedge product Grassmann ring symmetric product Representation of finite groups Complete reducibility Schur's lemma characters projection formulae Induced representation Frobenius reciprocity Representations of permutation groups
Quick Review of Rings and Ideals Ideals in quotient rings prime ideals under quotient existence of maximal ideals operations on ideals Chinese Remainder theorem nilradical and Jacobson radical extension and contraction of ideals under ring homomorphisms
Modules Free modules Projective Modules Tensor Product of Modules and Algebras Flat Faithfully Flat and Finitely Presented Modules Shanuel's Lemma
Localisation Local rings universal property of localisation extended and contracted ideals and prime ideals under localisation localisation and quotients exactness property Nagata's criterion for UFD and applications
Integral Extensions Integral closure normalisation and normal rings Cohen Seidenberg Going-Up Theorem Hilbert's Nullstellensatz and applications Introduction to Valuation rings Time permitting Introduction to Grobner basis
Algebraic Numbers and Algebraic Integers Brief review of integral extensions Norm trace and discriminant Existence of integral basis Dedekind domains ideal class group
Minkowski Theory Finiteness of class group Dirichlet unit theorem Factoring of prime ideals on extensions fundamental identity Quadratic number fields computation of class numbers prime decomposition Pell's equations
Hilbert's Ramification Theory Decomposition and inertia groups Cyclotomic fields Valuations completions local fields