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Divisibility and Irreducibility of Polynomials

The parallel between divisibility properties of the polynomials typeset structure over a field  typeset structure and the ring of integers typeset structureis 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 typeset structure is a ring we say that a polynomial typeset structure divides a polynomial typeset structure and we write typeset structure, provided typeset structure, for some polynomial typeset structure. We also say that typeset structure  is a divisor of typeset structure,  or that typeset structure is a factor of typeset structure.

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

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

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

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

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

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

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

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

This Theorem shows that the irreducible polynomials play in typeset structure the role of prime numbers in typeset structure. 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 typeset structure is irreducible in typeset structure if and only if typeset structure cannot be written as a product of two polynomials from typeset structure both having degrees less than the degree of typeset structure.

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

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

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

Uniqueness of the decomposition into irreducible polynomials: Let typeset structure be a field. Then every polynomial in typeset structure 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 typeset structure is delicate, and the answer heavily depends on the character of the underlying field typeset structure. The following result may be instrumental:

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

Cite this web-page as:

Štefan Porubský: Divisibility and Irreducibility of Polynomials.

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