yuchenglong {at} simis.cnThis course covers advanced topics in algebra. There are three parts I hope to cover. You may refer to course website of Math 371 Spring 2020 at UPenn in my personal page for related materials. Part I: Bilinar forms. Symmetric forms, Hermitian forms, and skew-symmetric forms. Orthogonality. Spectral Theorem. Conics and Quadrics. Key examples of classical groups, and their basic properties. Lie algebra (for such groups). Part II: Group representations. Irreducible representations and unitary representations. Characters. Schur’s Lemma. Modules over principal ideal domains. Free modules. Group rings. Noetherian rings. Structure of Abelian groups. Maschke's theorem. Constructions of representations, et cetera. Part III: Field extensions, algebraic extensions and algebraic closures, splitting fields, separable and inseparable extensions, Galois extensions, Galois correspondences, cyclotomic extensions, solvability by radicals, et cetera.