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Let K be a field of arbitrary characteristic. Let v:K->R union {infty} be defined by the following properties: 1. v(x)=infty<=>x=0, 2. v(xy)=v(x)+v(y) forall x,y in K, and 3. ...

The study of valuations which simplifies class field theory and the theory of function fields.

"Chaos" is a tricky thing to define. In fact, it is much easier to list properties that a system described as "chaotic" has rather than to give a precise definition of chaos. ...

A technical mathematical object defined in terms of a polynomial ring of n variables over a field k. Syzygies occur in tensors at rank 5, 7, 8, and all higher ranks, and play ...

Let F be a differential field with constant field K. For f in F, suppose that the equation g^'=f (i.e., g=intf) has a solution g in G, where G is an elementary extension of F ...

A number is said to be squarefree (or sometimes quadratfrei; Shanks 1993) if its prime decomposition contains no repeated factors. All primes are therefore trivially ...

The extension ring obtained from a commutative unit ring (other than the trivial ring) when allowing division by all non-zero divisors. The ring of fractions of an integral ...

An algorithm for determining the order of an elliptic curve E/F_p over the finite field F_p.

The Fibonacci numbers are the sequence of numbers {F_n}_(n=1)^infty defined by the linear recurrence equation F_n=F_(n-1)+F_(n-2) (1) with F_1=F_2=1. As a result of the ...

For any alpha in A (where A denotes the set of algebraic numbers), let |alpha|^_ denote the maximum of moduli of all conjugates of alpha. Then a function ...