BROWSE
ALPHABETICALLY
LEVEL:
Elementary
Advanced
Both
INCLUDE TOPICS:
Basic Math
Algebra
Analysis
Biography
Calculus
Comp Sci
Discrete
Economics
Foundations
Geometry
Graph Thry
History
Number Thry
Phys Sci
Statistics
Topology
Trigonometry
|
|
Diophantine equation – field isomorphism
Diophantine equation
A polynomial equation with integer coefficients. (Named after the 3rd century Greek mathematician Diophantus of Alexandria.)
 Cf. Hilbert's Problems (the tenth problem), Fermat's Last Theorem.
directed graph
A graph whose edges are directed, i.e. have distinguished ends. One end of every directed edge is called the head and the other is called the tail, and the edge is said to be from the tail to the head. In pictorial representations of graphs, directed edges are drawn to end with arrows, pointing to the head. The i, j entry in the adjacency matrix of a directed graph is the number of edges from vertex i to vertex j.
distributive
See distributive property.
distributive property
An algebraic property of numbers which states that for all numbers a, b, and c, a(b + c) = ab + ac.
Cf. commutative, associative.
divide
To divide a number a by another number b is to find a third number c such that the product of b and c is a, that is, b × c = a. The number a is called the dividend, the number b is called the divisor, and the number c is called the quotient. The operation of dividing may be denoted by a horizontal or diagonal slash separating the dividend and divisor (with the dividend on top), or by a horizontal dash with a dot above and below it placed between the dividend and divisor.
In the case of whole numbers a and b there may not be a whole number quotient; however, there are always unique whole numbers q and r such that a = b × q + r, with r < b. In this case q is called the quotient and r is called the remainder. If in a particular case r = 0, we say that b divides a, and this is often denoted by b|a.
dot product
See scalar product.
e
See Euler number.
epimorphism
A morphism f from X to Y is called an epimorphism when it is surjective, that is, when to each element y of Y there corresponds an x in X such that f(x) = y.
Cf. monomorphism.
Euclid number
See perfect number.
Euler number
The transcendental number e, approximately 2.71828.... It may be defined as the limit
or as the infinite sum
The function ex is called the exponential function, and has the property that it is its own derivative.
Euler phi function
Given a natural number n, this function returns the number of integers between 1 and n which are relatively prime to n.
factor
The factors of a natural number n are those whole numbers which divide it evenly. Since every number is divisible by itself and 1, neither 1 nor n is considered a proper factor of n.
More generally, if an expression can be written as a product of other expressions, then those other expressions are its factors. For instance, a polynomial with rational coefficients may always be factored into a product of first and/or second degree polynomial factors.
A number (or other expression) with no proper factors is called prime.
Graph Theory: A spanning subgraph of a given graph having at least one edge. In many contexts, it is interesting to determine whether some graph G can be decomposed into the edge-disjoint union of factors with some prescribed property. Such a decomposition is called a factorization of G. Often, the property in question is regularity of degree k. In this case, the factors are called k-factors, and the factorization a k-factorization. If G has a k-factorization, it is called k-factorable.
factorization
The process of factoring, that is, of finding proper factors.
Fermat’s Last Theorem
The Diophantine equation x n + y n = z n has no non-trivial solutions in integers x, y, and z for any n greater than 2. (There are infinitely many solutions in integers for n = 2, and these are called the Pythagorean triples.) Fermat penned this theorem in the margin of his copy of the Arithmetica of Diophantus of Alexandria, and added, “I have discovered a remarkable proof of this theorem, which unfortunately this margin is too small to contain.” He died without ever writing down the proof, and the theorem remained unproved for 300 years, until Andrew Wiles presented a proof at a Cambridge lecture in 1992.
Fermat’s Little Theorem
For any integer n and prime number p that does not divide n, n p - 1 is congruent to 1 modulo p.
Cf. Euler phi function.
Fermat test
A test for primality based on the converse of Fermat's little theorem:
the Fermat test for n base b is that the congruence bp-1  1 modulo n should hold. For prime n, this is always provided that
b and n are coprime; however there
are also some composite numbers which will pass the test. For example, 341 = 11 × 31 is composite, yet 2340 1 modulo 341, so we call 2 a Fermat pseudoprime to the base 2.
Even worse, Carmichael numbers are pseudoprimes for the Fermat test and any base.
Fortunately, pseudoprimes and Carmichael numbers are rare compared to primes.
Such tests are thus probabilistic, in that numbers passing the test are likely but not certain to be prime.
Fibonacci sequence
ARTICLE
The sequence discovered by the medieval mathematician Fibonacci, and described in his book Liber Abaci. See the article for a complete description.
field
A set together with two binary operations (called addition and multiplication) defined on its elements, and satisfying - 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).
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.
field automorphism
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.
field homomorphism
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.
field isomorphism
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.
|
|
 
|