Question

What is the highest degree of irreducible polynomial over Complex numbers ?

Answer 1

Answer:

In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field or ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as {\displaystyle \left(x-{\sqrt {2}}\right)\left(x+{\sqrt {2}}\right)}{\displaystyle \left(x-{\sqrt {2}}\right)\left(x+{\sqrt {2}}\right)} if it is considered as a polynomial with real coefficients. One says that the polynomial x2 − 2 is irreducible over the integers but not over the reals.

A polynomial that is irreducible over any field containing the coefficients is absolutely irreducible. By the fundamental theorem of algebra, a univariate polynomial is absolutely irreducible if and only if its degree is one. On the other hand, with several indeterminates, there are absolutely irreducible polynomials of any degree, such as {\displaystyle x^{2}+y^{n}-1,}{\displaystyle x^{2}+y^{n}-1,} for any positive integer n.

A polynomial that is not irreducible is sometimes said to be reducible.[1][2] However, this term must be used with care, as it may refer to other notions of reduction.

Irreducible polynomials appear naturally in the study of polynomial factorization and algebraic field extensions.

It is helpful to compare irreducible polynomials to prime numbers: prime numbers (together with the corresponding negative numbers of equal magnitude) are the irreducible integers. They exhibit many of the general properties of the concept of "irreducibility" that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors. When the coefficient ring is a field or other unique factorization domain, an irreducible polynomial is also called a prime polynomial, because it generates a prime ideal.

Answer 2

Concept:

A polynomial that cannot be factored into the product of two non-constant polynomials is known as an irreducible polynomial. The nature of the coefficients that are accepted for the potential factors, that is, the field to which the coefficients of the polynomial and its possible factors are thought to belong, determines irreducibility.

Solution:

If a polynomial of degree $n\geq 1$ with coefficients in a field F cannot be represented as a product of two non-constant polynomials of degree less than n over F, it is said to be irreducible over F.

Every degree one polynomial is irreducible.

Over R, the polynomial $x^2+1$ is irreducible, while over C, it is reducible.

The building blocks of all polynomials are irreducible polynomials.

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