A normal integral basis Theorem

Galois theory and the normal basis theorem

Generalizations of the Normal Basis Theorem of Finite Fields

We present a combinatorial characterization of sets of integers (rO,rl, ... ,TII-d, with OSri~"_2, " such that a'O,({1, ... ,a'·-1 fonn a basis of GF(q") over GF(q) for some ae GF'(q"). We use this characterization to prove the following generalization of the nonnal basis theorem fpr finite -fields of characteristic two: Let Ao.Al; ... ,A.,.-1 be integers in the range~<q, with at most one A.; e...

Cauchy Integral Theorem

where we use the notation dxI for (1.4) dxI = dxi1 ∧ dxi2 ∧ ... ∧ dxik for I = {i1, i2, ..., ik} with i1 < i2 < ... < ik. So ΩX is a free module over C ∞(X) generated by dxI . Obviously, Ω k X = 0 for k > n and ⊕ΩX is a graded ring (noncommutative without multiplicative identity) with multiplication defined by the wedge product (1.5) ∧ : (ω1, ω2)→ ω1 ∧ ω2. Note that (1.6) ω1 ∧ ω2 = (−1)12ω2 ∧ ω...

A Relative Integral Basis over Q( √−3) for the Normal Closure of a Pure Cubic Field

Let K be a pure cubic field. Let L be the normal closure of K. A relative integral basis (RIB) for L over Q(√ −3) is given. This RIB simplifies and completes the one given by Haghighi (1986).

A bifurcation theorem for noncoercive integral functionals

In this paper we study the existence of critical points for noncoercive functionals, whose principal part has a degenerate coerciveness. A bifurcation result at zero for the associated differential operator is established.