0

I'm auditing a number theory course. I'm trying to prove the following:

A polynomial of degree $n$ has at most $n$ roots

I know there are many proofs of this on SE, however none are particularly accessible (for me, at least). I know a bit of number theory, and a bit of abstract algebra. Any good starting points or hints?

K_M
  • 331
  • 4
    "None are particularly accessible" how come? Besides, what coefficients do those polynomials have? – Sarvesh Ravichandran Iyer Aug 03 '20 at 06:20
  • That's not the fundamental theorem of algebra. That theorem says that a complex polynomial has exactly $n$ roots (if we include multiplicity). So do you need a proof of the claim you wrote, or of the FTOA? – 5xum Aug 03 '20 at 06:21
  • My apologies. I need a proof of the claim as written. – K_M Aug 03 '20 at 06:22
  • 1
    I assume that the coefficients are drawn from a field. Argue by induction on the degree. The assertion plainly holds if the degree is 1. Assume the result for all polynomials of degree (n-1) as your induction hypothesis. Now consider a polynomial of degree n. If it has no roots there is nothing to prove. It it has a root, factor it out and use the induction hypothesis. – student Aug 03 '20 at 06:22
  • 1
    Does this answer your question? How do I prove that a polynomial F[x] of degree n has at most n roots – ShBh Aug 03 '20 at 06:30
  • You might need this https://en.m.wikipedia.org/wiki/Polynomial_remainder_theorem and the Unique Factorization Property of a polynomial ring over a field. – eduard Aug 03 '20 at 06:33
  • 2
    It will be difficult for anyone to answer this if you do not explain what troubles you have understanding those other proofs. –  Aug 03 '20 at 06:47
  • Do you know polynomial division? – Paul Frost Aug 03 '20 at 07:02

1 Answers1

3

Hint: There's a factor theorem saying that if $a$ is a root, then $p(x)$ has $x-a$ as a factor.