Field Extension by Galois Theory

Exercises in Field Theory and Galois Theory

1. Algebraic extensions (1) Let F be a finite field with characteristic p. Prove that |F | = p n for some n. (2) Using f (x) = x 2 + x − 1 and g(x) = x 3 − x + 1, construct finite fields containing 4, 8, 9, 27 elements. Write down multiplication tables for the fields with 4 and 9 elements and verify that the multiplicative groups of these fields are cyclic.

Quantum Theory on a Galois Field

Systems of free particles in a quantum theory based on a Galois field (GFQT) are discussed in detail. In this approach infinities cannot exist, the cosmological constant problem does not arise and one irreducible representation of the symmetry algebra necessarily describes a particle and its antiparticle simultaneously. As a consequence, well known results of the standard theory (spin-statistic...

Galois Theory for Arbitrary Field Extensions

Galois modules and class field theory Boas

10. Galois modules and class field theory Boas Erez In this section we shall try to present the reader with a sample of several significant instances where, on the way to proving results in Galois module theory, one is lead to use class field theory. Conversely, some contributions of Galois module theory to class fields theory are hinted at. We shall also single out some problems that in our op...

Coalgebra-galois Extensions from the Extension Theory Point of View

Coalgebra-Galois extensions generalise Hopf-Galois extensions, which can be viewed as non-commutative torsors. In this paper it is analysed when a coalgebra-Galois extension is a separable, split, or strongly separable extension.