In this chapter we return to several of the themes in Chapter 1.
We need to talk about the
greatest common divisor
of two polynomials,
rather than the
greatest common divisor
of two integers.
This will allow us to decide
when two polynomials are relatively prime.
The notion of a
prime number
is replaced by that of an
irreducible polynomial.
We can extend many of our results about
congruence class of integers
to
congruence classes of polynomials.
The point of saying this is that it will be worth your time
to review the definitions and theorems in Chapter 1.
In addition to generalizing ideas from the integers to polynomials, we want to go beyond high school algebra, to be able to work with coefficients that may not be real numbers. This motivates the definition of a field, which is quite closely related to the definition of a group (now there are two operations instead of just one). The point here is that you can also benefit from reviewing Chapter 3.
In this chapter we only study polynomials whose coefficients
belong to a field.
In Chapter 5 and Chapter 9 we will look at a more general situation,
where the coefficients may not be invertible.
The more general situation is related to early work
that was done in trying to solve Fermat's Last Theorem.
If you want to read some history at this point,
we recommend these two topics from
The MacTutor History of Mathematics
archive at the University of St. Andrews, in Scotland:
Quadratic, cubic and quartic equations and
Fermat's last theorem.
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