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Divisibility and Irreducibility of Polynomials

The parallel between divisibility properties of the polynomials over a field   and the ring of integers is based on the fact that both are examples of the so-called Euclidean rings. In the case of polynomials the "size" of polynomials is measured by their degree.

If is a ring we say that a polynomial divides a polynomial and we write , provided , for some polynomial . We also say that   is a divisor of ,  or that is a factor of .

The elements of the underlying ring can be viewed as the constant polynomials. Consequently the divisibility relation between constants viewed as the elements in coincides with divisibility relations between these elements viewed as polynomials. If the ring is unitary, we say that a polynomial is divisor of unity (in ), or a unit if there exists a polynomial   such that .

Two polynomials with a ring, are called associated (in or over ) provided   divides and simultaneously divides (with both divisibility relations understood within ). If two polynomials are associated then we write symbolically .

For instance, the polynomials and are not  associated in , but they are associated in .

Theorem. If , a field, are associated in then there exits a divisor of unity such that .

The divisibility relation in , a field, has the following properties:

• divisibility of polynomials if reflexive, i.e. for every ,
• if while and simultaneously , then also ,
• divisibility of polynomials is transitive, i.e. if   and then   for all .

If , R  unitary, then the following its divisors from are called trivial divisors:

• polynomials associated with unity ,
• polynomials associated with itself.

Polynomials having in , a ring, only trivial divisors are called irreducible (over )  or (seldom) prime (over ). If a polynomial is called reducible (over ) if it is not irreducible over .

Theorem. Let be a field. Then the following statements are equivalent:

• is irreducible,
• if with , then either or is a constant polynomial,
• if with , then either or .

This Theorem shows that the irreducible polynomials play in the role of prime numbers in . So for instance, the second and the third part of the previous Theorem enables us to
define equivalently irreducible polynomials similarly to prime numbers:

A polynomial is irreducible in if and only if cannot be written as a product of two polynomials from both having degrees less than the degree of .

Similarly as in the case of primes, also the irreducible polynomials over a field possess the following "prime" property:

Theorem. If is an irreducible polynomial over a field   and with , then either or in .

The next result (almost) completes the above mentioned parallelism between divisibility in and the set polynomials over a field . The analogue  of the Fundamental Theorem of Arithmetic says:

Uniqueness of the decomposition into irreducible polynomials: Let be a field. Then every polynomial in can be uniquely (i.e. up to the order of the factors in the product)  written as a product of irreducible polynomials.

The problem of characterization of irreducible polynomials over a given field is delicate, and the answer heavily depends on the character of the underlying field . The following result may be instrumental:

Theorem. If is irreducible over a field , then so is for every .

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