...This chapter works over a finite field k of odd characteristic. Fix an even integer 2n ≥ 4 and a monic polynomial f(x) ɛ k[x] of degree 2n, f ( x ) = ∑ i = 0 2...

...This chapter treats the case of characteristic two separately because it is somewhat simpler than the case of odd characteristic. Recall from the first paragraph of Chapter 25 that for k a finite field of characteristic 2, and any character χ of kˣ, the Tate...

...This chapter deals with the case of a finite field k, and a group scheme G/k. which is a form of ₘ. Euler characteristic middle convolution perverse sheaves quadratic extension Tannakian categories of perverse sheaves cohomology groups...

...The algebraic structure of fields is introduced and simple conlusions are draws from the field axioms. The notion of an ordering on fields is defined. Examples of fields, including finite fields, the rational and real numbers, and the complex numbers are discussed. Some basic features...

...Let k be a finite field, q its cardinality, p its characteristic, ≠ p a prime number, and G /k a smooth commutative group scheme which over k ¯ becomes isomorphic to k ¯ . This chapter...

...This chapter presents computational chemistry methods derived by taking derivatives of the approximate expressions for the energy of a molecule in the presence of electric or magnetic fields. It discusses the possibility of evaluating the derivatives numerically via the finite field approach...