Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2009-08-27 08:28:15

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Number of roots of polynomials


This can be proved using the equivalent of the division algorithm for polynomials over a field.



A simple proof of this using induction on the degree of

can be found on p.28 of D.J.H. Garling, A Course in Galois Theory (1986), Cambridge University Press.

Note that this applies to polynomials over a field only. This won’t work with rings in general, e.g.

and
in
. However, if
is an integral domain, we can apply the division algorithm to polynomials over its field of fractions
.

Let

be a polynomial of degree
in
. Let
be a root of
Considering
as a polynomial in
, we can write
where
and
. Then
.

Thus

. If
is another root of
distinct from
, write
where
and
. Then
as
.

Thus

. Continuing this way, we find that if
are all the distinct roots of
, then we have
for some
. It follows that
cannot have more than
roots in
since
has degree
and so cannot be factorized into more than
linear factors in
. And since a root of polynomial in
is also a root of a polynomial in
(Gauss’s lemma), our proof is complete. smile

If

is not an integral domain, it may be possible for an
th-degree polynomial in
to have more than
roots. For instance, as a polynomial over
, the quadratic polynomial
actually has 4 roots, namely 1, 3, 5, 7. That’s because
is not an integral domain and so we cannot construct its field of fractions.


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

Offline

#2 2009-08-28 11:49:08

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,677

Re: Number of roots of polynomials

Hi Jane;

Thanks for the reason why  x^2 + 7=0 mod 8 can have 4 solutions. I didn't know that.

Last edited by bobbym (2009-08-28 12:19:02)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

Board footer

Powered by FluxBB