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The group algebra K[G], where K is a field and G a group with the operation *, is the set of all linear combinations of finitely many elements of G with coefficients in K, ...

Let K be a field of arbitrary characteristic. Let v:K->R union {infty} be defined by the following properties: 1. v(x)=infty<=>x=0, 2. v(xy)=v(x)+v(y) forall x,y in K, and 3. ...

The name for the set of integers modulo m, denoted Z/mZ. If m is a prime p, then the modulus is a finite field F_p=Z/pZ.

Let (K,|·|) be a valuated field. The valuation group G is defined to be the set G={|x|:x in K,x!=0}, with the group operation being multiplication. It is a subgroup of the ...

Consider a first-order ODE in the slightly different form p(x,y)dx+q(x,y)dy=0. (1) Such an equation is said to be exact if (partialp)/(partialy)=(partialq)/(partialx). (2) ...

The transcendence degree of Q(pi), sometimes called the transcendental degree, is one because it is generated by one extra element. In contrast, Q(pi,pi^2) (which is the same ...

A three-dimensional surface with constant vector field on its boundary which traps at least one trajectory which enters it.

Gauge theory studies principal bundle connections, called gauge fields, on a principal bundle. These connections correspond to fields, in physics, such as an electromagnetic ...

If F is an algebraic Galois extension field of K such that the Galois group of the extension is Abelian, then F is said to be an Abelian extension of K. For example, ...

An algebraic variety is a generalization to n dimensions of algebraic curves. More technically, an algebraic variety is a reduced scheme of finite type over a field K. An ...