Results 1 to 2 of 2

Math Help - Ring of Polynomials - and their evaluation - What is the link?

  1. #1
    Super Member
    Joined
    Apr 2009
    Posts
    677

    Ring of Polynomials - and their evaluation - What is the link?

    Before I ask my question some standard definitions (or rather my understanding of these definitions):

    Let F[x] be a ring of polynomials over a field F.
    i.e. F[x] is a set of symbols of the form
    a_0x^0 + a_1x^1 + a_2x^2 + ... + a_nx^n where a_i \in F and x is just a symbol with no meaning whatsoever.

    To make it into a ring we define + and . for p(x),q(x) \in F[x]. These definitions are nothing but syntax rules to manipulate these symbols (using the + and . which are already defined for the Field, F)

    Under this definition we can write polynomial equations like
    p(x) = q(x).h(x) + r(x)
    Here all we mean is that when you apply the syntax rules for + and . defined on F[x] you end up with same symbols on both sides of the equation. We assign no meaning to these symbols.

    Now we define x as an element of some algebric stucture A where along with + and . we also define \alpha.x where \alpha \in F and x \in A. \alpha.x \in A. Under this, a polynomial evaluates into an element in the algebric structure A. (e.g. of such a structures are: field F itself or a vector space V over F)

    My question is why should the LHS and RHS of a polynomial equation (as presented above and others as well) evaluate to a same element in A ? What are the properties of F[x] and/or A that make this happen?

    Not sure if this is relevant question or something too trivial. But I guess it is better to ask. Any pointers would be welcome.
    Thanks
    Last edited by aman_cc; September 8th 2009 at 04:00 AM. Reason: additon
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    Posts
    677
    Quote Originally Posted by aman_cc View Post
    Before I ask my question some standard definitions (or rather my understanding of these definitions):

    Let F[x] be a ring of polynomials over a field F.
    i.e. F[x] is a set of symbols of the form
    a_0x^0 + a_1x^1 + a_2x^2 + ... + a_nx^n where a_i \in F and x is just a symbol with no meaning whatsoever.

    To make it into a ring we define + and . for p(x),q(x) \in F[x]. These definitions are nothing but syntax rules to manipulate these symbols (using the + and . which are already defined for the Field, F)

    Under this definition we can write polynomial equations like
    p(x) = q(x).h(x) + r(x)
    Here all we mean is that when you apply the syntax rules for + and . defined on F[x] you end up with same symbols on both sides of the equation. We assign no meaning to these symbols.

    Now we define x as an element of some algebric stucture A where along with + and . we also define \alpha.x where \alpha \in F and x \in A. \alpha.x \in A. Under this, a polynomial evaluates into an element in the algebric structure A. (e.g. of such a structures are: field F itself or a vector space V over F)

    My question is why should the LHS and RHS of a polynomial equation (as presented above and others as well) evaluate to a same element in A ? What are the properties of F[x] and/or A that make this happen?

    Not sure if this is relevant question or something too trivial. But I guess it is better to ask. Any pointers would be welcome.
    Thanks
    Requesting some help on this. Thanks
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Irreducible polynomials over ring of integers ?
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 28th 2011, 04:04 AM
  2. Replies: 10
    Last Post: February 12th 2011, 09:50 AM
  3. Ideals in polynomials ring
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 27th 2010, 07:21 PM
  4. Variable of ring of polynomials
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: February 24th 2010, 06:28 AM
  5. Ring of Polynomials
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 28th 2009, 09:11 AM

Search Tags


/mathhelpforum @mathhelpforum