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A group G is nilpotent if the upper central sequence 1=Z_0<=Z_1<=Z_2<=...<=Z_n<=... of the group terminates with Z_n=G for some n. Nilpotent groups have the property that ...

The set of nilpotent elements in a commutative ring is an ideal, and it is called the nilradical. Another equivalent description is that it is the intersection of the prime ...

Any cubic curve that passes through eight of the nine intersections of two given cubic curves automatically passes through the ninth.

If two curves phi and psi of multiplicities r_i!=0 and s_i!=0 have only ordinary points or ordinary singular points and cusps in common, then every curve which has at least ...

Any irreducible curve may be carried by a factorable Cremona transformation into one with none but ordinary singular points.

A group or other algebraic object is called non-Abelian if the law of commutativity does not always hold, i.e., if the object is not Abelian. For example, the group of ...

A normal extension is the splitting field for a collection of polynomials. In the case of a finite algebraic extension, only one polynomial is necessary.

Let G be a group with normal series (A_0, A_1, ..., A_r). A normal factor of G is a quotient group A_(k+1)/A_k for some index k<r. G is a solvable group iff all normal ...

In every residue class modulo p, there is exactly one integer polynomial with coefficients >=0 and <=p-1. This polynomial is called the normal polynomial modulo p in the ...

A normal series of a group G is a finite sequence (A_0,...,A_r) of normal subgroups such that I=A_0<|A_1<|...<|A_r=G.