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Informally, a field is a set, along with two operations defined on that set: an addition operation written as a + b, and a multiplication operation written as a ⋅ b, both of which behave similarly as they behave for rational numbers and real numbers, including the existence of an additive inverse −a for all elements a, and of a multiplicative inverse b −1 for every nonzero element b.
A non-zero element of an ordered field is infinitesimal if and only if its absolute value is smaller than any element of that is of the form , for a standard natural number. Ordered fields that have infinitesimal elements are also called non-Archimedean .
Field lines depicting the electric field created by a positive charge (left), negative charge (center), and uncharged object (right). A field line is a graphical visual aid for visualizing vector fields. It consists of an imaginary integral curve which is tangent to the field vector at each point along its length.
Over an algebraically closed field K (for example the complex numbers C), there are no finite-dimensional associative division algebras, except K itself. [2] Associative division algebras have no nonzero zero divisors. A finite-dimensional unital associative algebra (over any field) is a division algebra if and only if it has no nonzero zero ...
In mathematics, the Conway polynomial C p,n for the finite field F p n is a particular irreducible polynomial of degree n over F p that can be used to define a standard representation of F p n as a splitting field of C p,n. Conway polynomials were named after John H. Conway by Richard A. Parker, who was the first to define them and compute ...
The set of non-zero elements in GF(q) is an abelian group under the multiplication, of order q – 1. By Lagrange's theorem, there exists a divisor k of q – 1 such that x k = 1 for every non-zero x in GF(q). As the equation x k = 1 has at most k solutions in any field, q – 1 is the lowest possible value for k.
In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise it is a definite quadratic form. More explicitly, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if q(v) = 0.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
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