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Let be a field. An expression of the form

with and ,

is called a polynomial in one variable over .

The numbers are called the coefficients of the polynomial. Two polynomials are equal if all their coefficients coincide. The polynomials with for all are called constant polynomials, we write simply for them. In the zero polynomial, all coefficients equal . Using the sum symbol, one can write a polynomial briefly as .

The degree of a nonzero polynomial


is .

The zero polynomial does not have a degree. The coefficient , where is the degree of the polynomial, is called the leading coefficient of the polynomial. The term is called the leading term of the polynomial.

The set of all polynomials over a field is called polynomial ring over , it is denoted by , where is the variable of the polynomial ring.

Two polynomials

can be added by adding the components, i.e. the coefficients of the sum are just the sums of the coefficients of the two polynomials. In case , the "missing“ coefficients of are to be interpreted as . This addition is obviously associative and commutative, the zero polynomial is the neutral element and the negative polynomial is obtained by taking from every coefficient of its negative.

One can also multiply two polynomials, one puts

and extends this multiplication rule so that distributivity holds. This means one multiplies each summand with each summand and adds then everything together. One can describe this multiplication also explicitly by the rule:

For the degree the following rules hold.

The graph of a polynomial function from to of degree .

If a polynomial and an element are given, then one can insert into meaning that one replaces everywhere the variable by . This yields a mapping

which is called the corresponding polynomial function.

If and are polynomials, then the composition is described in the following way: one has to replace everywhere in the variable by and then simplify this expression. The result is again a polynomial. The order of and is important for this.