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Monic polynomial

Index Monic polynomial

In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. [1]

32 relations: Algebra, Algebraic equation, Algebraic integer, Algebraic number theory, Coefficient, Complex number, Complex quadratic polynomial, Constant function, Divisibility (ring theory), Field (mathematics), Field of fractions, Finite field, Gröbner basis, Indeterminate (variable), Integer, Integral domain, Integral element, Integrally closed domain, Irrational number, Irreducible polynomial, Monoid, Monomial order, Necklace (combinatorics), Partially ordered set, Polynomial, Polynomial ring, Prime number, Pseudorandom binary sequence, Rational number, Ring (mathematics), Semigroup, Unit (ring theory).

Algebra

Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.

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Algebraic equation

In mathematics, an algebraic equation or polynomial equation is an equation of the form where P and Q are polynomials with coefficients in some field, often the field of the rational numbers.

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Algebraic integer

In algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in (the set of integers).

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Algebraic number theory

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations.

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Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression.

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Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

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Complex quadratic polynomial

A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers.

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Constant function

In mathematics, a constant function is a function whose (output) value is the same for every input value.

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Divisibility (ring theory)

In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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Field of fractions

In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded.

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Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

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Gröbner basis

In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field.

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Indeterminate (variable)

In mathematics, and particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else but itself and is used as a placeholder in objects such as polynomials and formal power series.

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Integer

An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

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Integral domain

In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.

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Integral element

In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and a_j \in A such that That is to say, b is a root of a monic polynomial over A. If every element of B is integral over A, then it is said that B is integral over A, or equivalently B is an integral extension of A. If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial).

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Integrally closed domain

In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself.

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Irrational number

In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.

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Irreducible polynomial

In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials.

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Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.

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Monomial order

In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all (monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e.,.

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Necklace (combinatorics)

In combinatorics, a k-ary necklace of length n is an equivalence class of n-character strings over an alphabet of size k, taking all rotations as equivalent.

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Partially ordered set

In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

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Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

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Polynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.

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Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

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Pseudorandom binary sequence

A pseudorandom binary sequence (PRBS) is a binary sequence that, while generated with a deterministic algorithm, is difficult to predict and exhibits statistical behavior similar to a truly random sequence.

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Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

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Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

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Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.

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Unit (ring theory)

In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.

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References

[1] https://en.wikipedia.org/wiki/Monic_polynomial

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