Turn on thread page Beta
    • Thread Starter
    Offline

    1
    ReputationRep:
    I was wondering about this: are there always finitely many polynomials (up to scalar multiples) of a fixed degree (say n) such that a complex number z is a root of these polynomials?

    Maybe this is obvious?
    Offline

    2
    ReputationRep:
    Could you clarify - from what I can make out, it seems obvious, but perhaps I'm misunderstanding you - clearly the number of polynomials of degree (n-1) isn't finite. Just multiply any of these by (w-z) to get a poly of degree n which has z as a root.
    Offline

    12
    ReputationRep:
    Suppose there are finite polys P(n) which has z is one of their roots.
    Then
    Let P(x) = {P1, P2, ....} be set of all poly P(n) which has root z.
    Let X = {z, z1, z2, ....,} be the set of all the roots of all the polys P(n)
    As number of P(n) is finite, and each P(n) has at most n roots, then X is finite set.
    -> Suppose z* is not in Z.
    Take P(w) = (w - z)n-1(w - z*).
    So P(w) has degree of n, has 1 root is z, but P(w) is not in P(x)
    -> contradiction

    well, I don't see it right
    Offline

    1
    ReputationRep:
    (Original post by Wrangler)
    Actually, re-reading your post, I don't think that's what your asking! Could you clarify - from what I can make out, it seems obvious, but perhaps I'm misunderstanding you - clearly the number of polynomials of degree (n-1) isn't finite. Just multiply any of these by (w-z) to get a poly of degree n which has z as a root.
    Yes that's all I could make of the question, in which case it's obviously not so. :confused:
    • Thread Starter
    Offline

    1
    ReputationRep:
    Well, I was showing that rt(3)+rt(-5) is algebraic using a deg 4 polynomial. That is easy. But I was wondering how many polynomials other than the one I found would work to show that its algebraic--i.e. for how many polynomials of deg 4 (say) is rt(3)+rt(-5) a root?
    Offline

    10
    ReputationRep:
    Am I just horribly over simplfying this but couldn't you just think about it like :

    P(x) = (x-a)(x-b)(x-c)(x-d)

    Let a = \sqrt{3}+i\sqrt{5}, b = \sqrt{3}-i\sqrt{5} then since you are free to pick c and d to be any real numbers or pair of complex conjugate roots there are uncountably infinite different 4th degree polynomials whose coefficents are real but who have \sqrt{3}+i\sqrt{5} as a root.
    Offline

    1
    ReputationRep:
    (Original post by J.F.N)
    Well, I was showing that rt(3)+rt(-5) is algebraic using a deg 4 polynomial. That is easy. But I was wondering how many polynomials other than the one I found would work to show that its algebraic--i.e. for how many polynomials of deg 4 (say) is rt(3)+rt(-5) a root?
    You are talking about minimal polynomials I think. Given an algebraic number z there is a unique monic polynomial of least degree which has z as a root. This poly is called its minimal polynomial. Of necessity it is irreducible.
    • Thread Starter
    Offline

    1
    ReputationRep:
    (Original post by Neapolitan)
    You are talking about minimal polynomials I think. Given an algebraic number z there is a unique monic polynomial of least degree which has z as a root. This poly is called its minimal polynomial. Of necessity it is irreducible.
    Irreducible over Q? Why is that obvious?
    Offline

    10
    ReputationRep:
    If it weren't then we can factor it and one of its factors would have z as a root yet have degree less than the minimal polynomial.
    • Thread Starter
    Offline

    1
    ReputationRep:
    (Original post by dvs)
    If it weren't then we can factor it and one of its factors would have z as a root yet have degree less than the minimal polynomial.
    I see.

    Also, another question: if F is a field, then the units in the polynomial ring F[x] are just all non-zero constant elements of the field, right?
    Offline

    10
    ReputationRep:
    Yeah
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: November 26, 2005

University open days

  • University of East Anglia
    UEA Mini Open Day Undergraduate
    Fri, 23 Nov '18
  • Norwich University of the Arts
    Undergraduate Open Days Undergraduate
    Fri, 23 Nov '18
  • Edge Hill University
    All Faculties Undergraduate
    Sat, 24 Nov '18
Poll
Black Friday: Yay or Nay?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Equations

Best calculators for A level Maths

Tips on which model to get

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.