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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, ...

A polynomial is said to be irreducible if it cannot be factored into nontrivial polynomials over the same field. For example, in the field of rational polynomials Q[x] (i.e., ...

For a real number x in (0,1), let m be the number of terms in the convergent to a regular continued fraction that are required to represent n decimal places of x. Then for ...

The smallest positive composite number and the first even perfect square. Four is the smallest even number appearing in a Pythagorean triple: 3, 4, 5. In the numerology of ...

Let K be a number field, then each fractional ideal I of K belongs to an equivalence class [I] consisting of all fractional ideals J satisfying I=alphaJ for some nonzero ...

A Gaussian integer is a complex number a+bi where a and b are integers. The Gaussian integers are members of the imaginary quadratic field Q(sqrt(-1)) and form a ring often ...

A vector basis of a vector space V is defined as a subset v_1,...,v_n of vectors in V that are linearly independent and span V. Consequently, if (v_1,v_2,...,v_n) is a list ...

An algorithm that can always be used to decide whether a given polynomial is free of zeros in the closed unit disk (or, using an entire linear transformation, to any other ...

Let E be an elliptic curve defined over the field of rationals Q(sqrt(-d)) having equation y^2=x^3+ax+b with a and b integers. Let P be a point on E with integer coordinates ...

Consider h_+(d) proper equivalence classes of forms with discriminant d equal to the field discriminant, then they can be subdivided equally into 2^(r-1) genera of ...