Roots of Polynomials II
This one might have to suffice for two days. The movers are almost done loading the truck here in New Orleans, and I’ll be taking today and tomorrow to drive to Maryland.
We can actually tease out more information from the factorization we constructed yesterday. Bur first we need a little definition.
Remember that when we set up the algebra of polynomials we noted that the coefficients have to be all zero after some finite number of them. Thus there must be a greatest nonzero coefficient . The index
corresponding to this coefficient we call the “degree” of the polynomial. If all the coefficients are zero — giving the zero polynomial — then a common convention is to assign it degree
. This actually isn’t completely arbitrary, but the details won’t concern us until later.
So, armed with this information, look at how we constructed the factorization . We replaced each term
in
with the term
. Then we factored out
from this term, giving
. So the highest power of
that shows up in this term (with a nonzero coefficient) is
. And the highest power coming from all the terms of
will be
. The power
shows up only once in the expression for
, so there’s no way for two such terms to add together and make its coefficient turn out to be zero, and no higher power of
ever shows up at all. Thus the degree of
is one less than that of
.
So what does this gain us? Well, each time we find a root we can factor out a term like , which reduces the degree by
. So if
has degree
there can only be at most
roots!
A nonzero constant polynomial has degree
, but it also has no roots! Perfect.
A linear polynomial has degree
, and it has exactly one root:
.
Now let’s assume that our statement is true for all polynomials of degree : they have
or fewer roots. Then given a polynomial
of degree
either
has a root
or it doesn’t. If it doesn’t, then we’re already done. If it does, then we can factor
, where
has degree
. But then
can have at most
roots, and thus
can have at most
roots!
A nice little corollary of this is that if our base field is infinite (like it is for the most familiar examples) then only the zero polynomial can give us the zero function when we evaluate it at various field elements. That is, if
for all
, then
. This must be true because
has an infinite number of roots, and so no finite degree polynomial can possibly have that many roots. The only possibility left is the zero polynomial.
Just to be clear, though, let’s look at this one counterexample. Think about the field we used when we talked about Set. The polynomial
is not the zero polynomial, but
is the zero function. Indeed
,
, and
, which is divisible by
, and so is the same as
in this field.