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This module builds on previous knowledge and gives an in depth understanding of Rings and Fields, covering the following topics. Field Extensions. Fields as factor rings. Algebraic elements, minimal polynomials, existence of roots of polynomials. Existence and uniqueness of splitting fields. The impossibility proofs: squaring the circle, duplicating the cube, trisecting an angle. Finite Fields. Fields of rational functions. Theory of a single linear transformation. Jordan normal form. Cayley-Hamilton theorem.
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