field group automorphism
A set together with two binary operations (called addition and multiplication) defined on its elements, and satisfying
In other words, a field is a commutative division ring. If in addition the field is linearly ordered, it is called an ordered field. An ordered field is complete if every subset of the field with an upper bound has a least upper bound. The real numbers are fully characterized by the fact that they form a complete ordered field, that is, every complete ordered field is isomorphic to the real numbers.
- the set is an Abelian group under the addition operation,
- the set is a commutative ring with addition as the group operation and multiplication as the ring operation, and
- every element except the additive identity has a multiplicative inverse (i.e., the set is a division ring).
A field isomorphism from a field to itself. That is, a bijective function from a field to itself that preserves addition and multiplication, i.e. f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b).
Cf. field isomorphism.
A function from one field to another that preserves addition and multiplication, i.e. f(a+b) = f(a) + f(b) and f(ab) = f(a)f(b). All field homomorphisms are actually isomorphisms of the domain with some subfield of the range.
Cf. field isomorphism, field automorphism.
A field homomorphism that is both “one-to-one” and “onto,” that is, a bijective function from one field to another that preserves addition and multiplication, i.e. f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b).
Cf. field automorphism.
General: A drawing, picture, or illustration, usually accompanying a text description. Also synonymous with digit or numeral.
Geometry: A graphical (visual) representation of points, lines, curves, surfaces, solids, or regions. The word “figure” may also be used to refer to the abstract object or set of points thus represented.
Cf. plane figure.
A graph whose vertex and edge sets are finite. In all papers, theorems, conjectures, textbooks, discussions, and interactions of graph theory all graphs considered are always finite, unless explicitly specified otherwise.
Cf. infinite graph.
A set X is finite if there is a natural number n (possibly 0) such that X can be said to have exactly n elements. More formally, a set is finite if it is not bijective with any proper subset of itself.
Given a binary relation R on sets A and B, we say that the R is a function if each element of A is paired with at most one element of B. In this case we call A the domain set, and we call B the range set. If f is such a function and the ordered pair (x,y) is an element of the function, we typically write f(x) = y.
More generally, if R is an n-place relation, then R is a function of n-1 variables if each (n-1)-tuple is matched with at most one element of the range set (i.e., the nth set of the Cartesian product on which the relation is defined). If in addition there is at most one element of the domain set matched with any given element of the range set, then the function is called “one-to-one” or injective. If the function maps at least one element of the domain set to every element of its range set, then it is called “onto” or surjective. A function which is both injective and surjective is called bijective. Functions with range in the real numbers or complex numbers are called real-valued or complex-valued respectively.
Fundamental Theorem of Algebra
The field of complex numbers is
algebraically closed, that is, any polynomial with complex coefficients has a complex root.
Fundamental Theorem of Arithmetic
Every natural number is either prime or may be decomposed uniquely into prime factors.
Abbreviation for greatest common divisor.
A curve on a space is a geodesic if, given any two points on the curve, the segment of the curve joining the points is the shortest such curve. In Euclidean space, geodesics are straight lines. On a sphere, the geodesics are the great circles of the sphere.
Graph Theory: A shortest path between two vertices is sometimes called a geodesic.
Abbreviation for greatest lower bound.
General: A picture or diagram representing one or more relationships among certain objects or quantities.
Functions: The collection of ordered tuples (x,y), consisting of domain elements x (possibly also tuples) and corresponding range elements y. Although informally we often mean the picture when we speak of the graph of a function, the graph should always be formally thought of as a set of tuples.
Graph Theory: A collection of nodes (also called vertices), together with a set of edges joining the nodes. Formally, a graph G is a set of vertices V, a set of edges E, and a symmetric relation between them specifying incidence. Every edge must be incident on exactly two (not necessarily distinct) vertices, called its ends. Two vertices incident on the same edge are called adjacent. Graphs are often represented pictorially, with the vertices being points (or small circles or squares or some such), and the edges being curves (often line segments), with the endpoints of each such curve being the ends of that edge in the graph.
Cf. Konigsberg Bridge Problem.
On a sphere, a great circle is a circle containing the endpoints of a diameter of the sphere, that is, a circle of maximal size. A geodesic on a sphere always lies on a great circle.
greatest common divisor
The greatest common divisor of two natural numbers a and b is the largest natural number c such that c divides a and c divides b.
greatest lower bound
A lower bound which is greater than or equal to every other lower bound.
Cf. Least upper bound.
Letters of the Greek alphabet are commonly used in mathematics. See the article for a full description.
A set together with a binary operation defined on its elements, satisfying
If the group operation is commutative, the group is called Abelian. If the condition that every element have an inverse is dropped, the set is called a monoid. If in addition the requirement that an identity element exist is dropped, the set is called a semigroup. Finally, if the condition that the group operation be associative is dropped, the set is called a groupoid.
- The group operation is associative,
- There is an identity element, i.e., an element e of the group so that for any other element a of the group we have ea = ae = a, and
- for every element a of the group there is an element a´, called the inverse of a, so that aa´ = a´a = e.
Cf. field, module, ring
A group isomorphism from a group to itself. That is, a bijective function from a group to itself that preserves the group operation, i.e. f(ab) = f(a)f(b) for all a, b.
Cf. group homomorphism, group isomorphism.