ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

Example 57: Polynomials with real coefficients (see nr 120)
The set of real functions on R is a ring.
The set of polynomials is a subring and so a ring. This ring is usually denoted
by R[X]. In the sequel we shall denote it P.
In P we have a lot of results which are analogous to results valid in the ring of
integers when we replace the order of the integers by the order of polynomials
according to degree: one polynomial is considered to be less than another one
if its degree is less :
You can carry out division with remainder. A well known algorithm learns you
how to. The resulting remainder will be less than the divisor. And so you can
use a Euclidean algorithm to find a greatest common divisor. And you can
define an extended algorithm: to any two given polynomials a and b you can
express their greatest common divisor c in the form xa + yb, where x and y are
polynomials.
Let p be a polynomial. Put I = pP, then I is an ideal consisting of all multiples
of p. Therefore P/I is a ring, the ring of remainder classes modulo p.
Lets see how multiplication works:
[a + bx][c + dx] = [(a + bx)(c + dx)] = [ac + (bc + ad)x + bdx 2 ] = [ac − bd + (bc + ad)x +
[ac − bd + (ad − bc]x] + [bd(1 + x 2 )] = [ac − bd + (ad − bc]x]
From which we see that the mapping (a+ib) → [a+bx] is a homomorphism from
the complex numbers with multiplication. It is actually a field isomorphism.
If q divides p then I = pP must be a subideal of J = qP. If moreover p not
divides q then the subideal is a proper subideal.
The polynomial p is said to be irreducible if it can not be factored into two
polynomials, unless one of the factors is a constant. This is the polynomial
analogue of a prime number.
If p is irreducible then I is a maximal ideal (show it !) and the quotient ring
consequently a field.
The polynomial p(x) = 1 + x 2 is irreducible and the associated field is isomorphic
with the field of complex numbers.
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