Galois Theory: Extensions and Residual Fields

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Page 3 of 4

Explores ramification theory, residue fields, Galois extensions, and decomposition groups in algebraic number theory.

Explores Frobenius theorems in number theory, ideal class groups, norm properties, and geometry of numbers.

Explores the definition of Q_p and the completion of Q,1(p) to form Qp.

Covers matrix calculations, basis change, field extensions, complex numbers modulus, and polar decomposition.

Revisits the Galoisian correspondence and explores group actions in intermediate coatings.

Introduces Galois fields, elliptic curves, factorization algorithms, and the discrete logarithm problem in cryptography.

Explores the properties and applications of finite fields, including isomorphism and cyclic properties.

Introduces algebras over a field, k-linear endomorphisms, and commutative algebras.

Covers homomorphisms, algebraic extensions, cutting, splitting, and separable elements in Fok [x].

Explores separable extensions and Dedekind rings, focusing on coefficients and prime ideals.