Abel's theorem in problems and solutions - School of Mathematics

48 Chapter 2
If K is a field then it is possible, as for the field of the real numbers,
to consider the polynomials with coefficients in the field K, or, in other
words, the polynomials over K.
DEFINITION. An expression like ( being a natural number)
where are elements of the field K, and is called a
polynomial of degree in one variable over K.
If is an element of the field K the expression is itself considered
as a polynomial over K, and if it represents a polynomial of degree
zero, whereas if the degree of this polynomial is considered to be
undefined.
The elements are called the coefficients of the polynomial
(2.1) and the leading coefficient.
Two polynomials in one variable are considered to be equal if and
only if the coefficients of the terms of the same degree in both polynomials
coincide. Let
If in the second member of this equation one replaces with an element
of the field K and one carries out the calculations indicated, i.e., the
operations of addition and multiplication in the field K, one obtains as
a result some element of the field K. One thus writes If
where 0 is the zero element of field K, one says that is a root
of the equation one also says that is a root of the polynomial
The polynomials on any field can be added, subtracted, and multiplied.
The sum of two polynomials and is a polynomial in
which the coefficient of is equal to the sum (in the
field K) of the coefficients of in the polynomials and In
the same way one defines the difference of two polynomials. It is evident
that the degree of the sum or of the difference of two polynomials is not
higher than the maximum of the degree of the given polynomials.
To calculate the product of the polynomials and one must
multiply every monomial of the polynomial by every monomial
of the polynomial according to the rule