Polynomial ring

When a commutative ring having a, then the polynomial is the set of polynomials with coefficients in the ring and the variables with the usual addition and multiplication of polynomials. Must be distinguished from the abstract algebra, the polynomial functions, not least because different polynomials can induce the same polynomial function.

  • 2.1 degrees set
  • 2.2 Elementary operations, polynomial algebra
  • 2.3 homomorphisms
  • 2.4 Algebraic properties
  • 3.1 Examples
  • 4.1 A polynomial over a finite field
  • 4.2 A polynomial with two variables
  • 4.3 polynomials in the complex domain

Definitions

The polynomial ring R [ X]

Is the set of all finite sequences

Where " almost all " means " for all but finitely many".

The addition is componentwise performed:

And the convolution of the sequences defined the multiplication

Through these links is defined on the space of finite sequences, a ring structure, said ring is referred to as.

This ring is defined as

From the definition of multiplication by convolution it follows that ( k times ) has the unit element are exactly at the point - s, otherwise there is a sequence of all zeros.

With the generator now, each element of clearly common in the Polynomschreibweise

Are shown.

If a further ring expansion of and, then, by the Einsetzungshomomorphismus

Be defined. For the element to write well.

Thus we obtain the polynomial ring over the indeterminates. The individual sequence elements are called the coefficients of the polynomial.

The polynomial ring in several variables

The polynomial ring in several variables is recursively defined by:

So you discussed here are polynomials in the variables with coefficients in the polynomial ring, and this is again defined the same. This can be as long as continue until you arrive at the definition of the polynomial ring in one variable. In each element can be clearly identified as

. Write

Polynomial ring in any number of unknowns ( with an index set ) can be defined via the free commutative monoid over or as the colimit of polynomial rings over finite subsets of either the Monoidring.

The quotient field

If a body is, so is the name of the quotient field of the rational function field. Analogue of the quotient field of a polynomial ring over several indeterminates is denoted by.

Properties

Degree set

Under the degree of a polynomial, is defined as the number

It applies to all

  • .

For this degree theorem follows in particular that if a body is, the units nonzero correspond exactly to the constant polynomials with degree zero.

Wherein a body is defined by the membership function to a Euclidean ring: there is a division with remainder, in which the residue or a lesser degree than the divisor.

Elementary operations, polynomial algebra

In the Polynomschreibweise see addition and multiplication for elements and of the polynomial as follows:

The polynomial ring is not just a commutative ring, but also a module, where the scalar multiplication is defined componentwise. This is actually a commutative associative algebra.

Homomorphisms

If and commutative rings with and are a homomorphism, then also

If and commutative rings with and are a homomorphism, then there is for each a unique homomorphism, which is restricted to the same and applies to the, namely.

Algebraic properties

Is a commutative ring with, then:

  • If no zero divisors, as well.
  • Is factorial, so also ( Lemma of Gauss )
  • If a body is, so is Euclidean and hence a principal ideal ring.
  • Is a Noetherian, then for the dimension of the polynomial ring in one variable over:
  • Is noetherian, then the polynomial ring with coefficients in noetherian. ( Hilbert basis theorem )
  • Is an integral domain and so has at most zeros. This is wrong on non- integrity rings in general.
  • A polynomial is invertible if and only in, if is invertible and all other coefficients are nilpotent in. In particular, a polynomial is an integral domain invertible if there is a constant polynomial, where one unit is.

Polynomial function and Einsetzungshomomorphismus

Is

A polynomial, so called

To the associated polynomial function. General also defined for each [ [ ring ( algebra) # lower and upper ring | extension] ] by a polynomial function, the index is often omitted.

Conversely, there is a fixed element of a commutative extension of a polynomial with variable ring homomorphism

Or

The evaluation ( - shomomorphismus ) or for setting ( - shomomorphismus ) is called from.

Examples

  • If we set and, as is the identity map; .
  • Consider a polynomial with additional indeterminates (see polynomials with several variables ) as an extension of, then, analogous to the construction of the previous example as a monomorphism of Einsetzungshomomorphismus in

Examples

A polynomial over a finite field

As in the finite field, the device group is cyclic with the order applies to the equation. This is why the polynomial function of the polynomial

The zero function, although it is not the zero polynomial.

Is a prime number, then this corresponds exactly to the small Fermat's theorem.

A polynomial with two variables

Be. The real roots of this polynomial are all points of the unit circle, in formulas

So there is an infinite number of zeros, unlike in or where each polynomial has only finitely many zeros.

Polynomials in the complex domain

Every complex polynomial of degree has exactly zeros in, if you count each zero according to their multiplicity. This is called a zero -fold, if a divisor is, however, no longer.

In particular, this fundamental theorem of algebra is also true for real polynomials, if one conceives these as polynomials in. For example, the polynomial has the zeros and, as well, and so true.

Unit (ring theory) Unique factorization domain Krulldimension Hilbert's basis theorem Serge Lang
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