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A

**field**which is complete with respect to a discrete valuation is called a local**field**if its**field**of residue classes is finite. The Hasse principle is one of the chief ...A quadratic

**field**Q(sqrt(D)) with D>0.A

**field**K is said to be an extension**field**(or**field**extension, or extension), denoted K/F, of a**field**F if F is a subfield of K. For example, the complex numbers are an ...In a local ring R, there is only one maximal ideal m. Hence, R has only one quotient ring R/m which is a

**field**. This**field**is called the residue**field**.The order of a finite

**field**is the number of elements it contains.A vector

**field**v for which the curl vanishes, del xv=0.A place nu of a number

**field**k is an isomorphism class of**field**maps k onto a dense subfield of a nondiscrete locally compact**field**k_nu. In the function**field**case, let F be ...If r is an algebraic number of degree n, then the totality of all expressions that can be constructed from r by repeated additions, subtractions, multiplications, and ...

A finite

**field**is a**field**with a finite**field**order (i.e., number of elements), also called a Galois**field**. The order of a finite**field**is always a prime or a power of a ...A vector

**field**u satisfying the vector identity ux(del xu)=0 where AxB is the cross product and del xA is the curl is said to be a Beltrami**field**....