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Covers the Jordan Normal Form and the identification of torsion blocks within matrices.
Covers the algebraic closure of Qp and the definition of p-adic complex numbers, exploring roots' continuous dependence on coefficients.
Covers the proof of Hensel's Lemma and a review of field theory, including Newton's approximation and p-adic complex numbers.
Explores Galois theory fundamentals, including separable elements, decomposition fields, and Galois groups, emphasizing the importance of finite degree extensions and the structure of Galois extensions.
Covers normed spaces, dual spaces, Banach spaces, Hilbert spaces, weak and strong convergence, reflexive spaces, and the Hahn-Banach theorem.
Covers the concept of finite degree extensions in Galois theory, focusing on separable extensions.
Covers the structure of finite dimensional algebras and the characterization of semisimple algebras.
Covers affine algebraic sets, hypersurfaces, elliptic curves, ideals, and Noetherian rings in algebraic geometry.
