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Eisenstein's irreducibility criterion is a sufficient condition assuring that an integer polynomial p(x) is irreducible in the polynomial ring Q[x]. The polynomial ...

The dimension d of any irreducible representation of a group G must be a divisor of the index of each maximal normal Abelian subgroup of G. Note that while Itô's theorem was ...

Suppose that V is a group representation of G, and W is a group representation of H. Then the vector space tensor product V tensor W is a group representation of the group ...

A representation of a Lie algebra g is a linear transformation psi:g->M(V), where M(V) is the set of all linear transformations of a vector space V. In particular, if V=R^n, ...

In an integral domain R, the decomposition of a nonzero noninvertible element a as a product of prime (or irreducible) factors a=p_1...p_n, (1) is unique if every other ...

While many computations admit shortcuts that allow them to be performed more rapidly, others cannot be sped up. Computations that cannot be sped up by means of any shortcut ...

The Farey sequence F_n for any positive integer n is the set of irreducible rational numbers a/b with 0<=a<=b<=n and (a,b)=1 arranged in increasing order. The first few are ...

For an atomic integral domain R (i.e., one in which every nonzero nonunit can be factored as a product of irreducible elements) with I(R) the set of irreducible elements, the ...

A separable algebraic extension E of F for which every irreducible polynomial in F which has a single root in E has all its roots in E is said to be Galoisian. Galoisian ...

There are at least two statements known as Schur's lemma. 1. The endomorphism ring of an irreducible module is a division algebra. 2. Let V, W be irreducible (linear) ...