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The set of all nonsingular affine transformations of a translation in space constitutes a group known as the affine group. The affine group contains the full linear group and ...

The field F^_ is called an algebraic closure of F if F^_ is algebraic over F and if every polynomial f(x) in F[x] splits completely over F^_, so that F^_ can be said to ...

A field K is said to be algebraically closed if every polynomial with coefficients in K has a root in K.

Three elements x, y and z of a set S are said to be associative under a binary operation * if they satisfy x*(y*z)=(x*y)*z. (1) Real numbers are associative under addition ...

The field of complex numbers, denoted C.

If P(x) is an irreducible cubic polynomial all of whose roots are real, then to obtain them by radicals, you must take roots of nonreal numbers at some point.

Two elements x and y of a set S are said to be commutative under a binary operation * if they satisfy x*y=y*x. (1) Real numbers are commutative under addition x+y=y+x (2) and ...

Discrete group theory is a broad subject covering certain aspects of groups. Such topics as free groups, group presentations, fundamental groups, Kleinian groups, and ...

The degree (or relative degree, or index) of an extension field K/F, denoted [K:F], is the dimension of K as a vector space over F, i.e., [K:F]=dim_FK. If [K:F] is finite, ...

Let R be a ring. If phi:R->S is a ring homomorphism, then Ker(phi) is an ideal of R, phi(R) is a subring of S, and R/Ker(phi)=phi(R).