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The simplest algebraic language, denoted D. If X is the alphabet {x,x^_}, then D is the set of words u of X which satisfy 1. |u|_x=|u|_(x^_), where |u|_x is the numbers of ...

The term external direct product is used to refer to either the external direct sum of groups under the group operation of multiplication, or over infinitely many spaces in ...

Since |(a+ib)(c+id)| = |a+ib||c+di| (1) |(ac-bd)+i(bc+ad)| = sqrt(a^2+b^2)sqrt(c^2+d^2), (2) it follows that (a^2+b^2)(c^2+d^2) = (ac-bd)^2+(bc+ad)^2 (3) = e^2+f^2. (4) This ...

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 ...

For even h, (1) (Nagell 1951, p. 176). Writing out symbolically, sum_(n=0)^h((-1)^nproduct_(k=0)^(n-1)(1-x^(h-k)))/(product_(k=1)^(n)(1-x^k))=product_(k=0)^(h/2-1)1-x^(2k+1), ...

An algebraic ring which appears in treatments of duality in algebraic geometry. Let A be a local Artinian ring with m subset A its maximal ideal. Then A is a Gorenstein ring ...

An n-dimensional manifold M is said to be a homotopy sphere, if it is homotopy equivalent to the n-sphere S^n. Thus no homotopy group can distinguish between M and S^n. The ...

The ideal quotient (a:b) is an analog of division for ideals in a commutative ring R, (a:b)={x in R:xb subset a}. The ideal quotient is always another ideal. However, this ...

A special ideal in a commutative ring R. The Jacobson radical is the intersection of the maximal ideals in R. It could be the zero ideal, as in the case of the integers.

This is proven in Rademacher and Toeplitz (1957).