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An algebraic function is a function f(x) which satisfies p(x,f(x))=0, where p(x,y) is a polynomial in x and y with integer coefficients. Functions that can be constructed ...

A cyclotomic field Q(zeta) is obtained by adjoining a primitive root of unity zeta, say zeta^n=1, to the rational numbers Q. Since zeta is primitive, zeta^k is also an nth ...

A fractal curve created from the base curve and motif illustrated above (Lauwerier 1991, p. 37). As illustrated above, the number of segments after the nth iteration is ...

An important result in valuation theory which gives information on finding roots of polynomials. Hensel's lemma is formally stated as follows. Let (K,|·|) be a complete ...

The ten-of-diamonds decahedron is a stereohedron and space-filling polyhedron on 8 vertices, 16 edges, and 10 faces (8 of which are non-equilateral triangles and two of which ...

An algebraic integer of the form a+bsqrt(D) where D is squarefree forms a quadratic field and is denoted Q(sqrt(D)). If D>0, the field is called a real quadratic field, and ...

There are many unsolved problems in mathematics. Several famous problems which have recently been solved include: 1. The Pólya conjecture (disproven by Haselgrove 1958, ...

A generalization of the p-adic norm first proposed by Kürschák in 1913. A valuation |·| on a field K is a function from K to the real numbers R such that the following ...

If r is a root of a nonzero polynomial equation a_nx^n+a_(n-1)x^(n-1)+...+a_1x+a_0=0, (1) where the a_is are integers (or equivalently, rational numbers) and r satisfies no ...

A p-adic number is an extension of the field of rationals such that congruences modulo powers of a fixed prime p are related to proximity in the so called "p-adic metric." ...