Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Polynomials over a ring evaluated at a value?

  1. Apr 30, 2015 #1
    In ring theory, a polynomial over a rings, say ## R[x] ## is presented as an abstract object of the form:
    ## p(x) = a_{n}x^{n} + ...+ a_{1}x + a_{0} ## where the coefficients ## a_{n}...a_{0} ## are from a ring R with unity and ##x## is a formal symbol.

    So what is the significance of ## p(x+1) ## ? In a high school algebra, one would simply interpret this as ##p(x)## with every instance of the variable ##x## replaced by ##x+1##. But in this notation, what does ## x + 1 ## even mean? It is itself a one-degree polynomial of ## R[x]## but then what does the notation ##p(x+1)## refer to?

    BiP
     
  2. jcsd
  3. May 1, 2015 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    It means exactly what it appears to mean. Since R is a ring with unity, "1" is that unity (multiplicative identity) and x+ 1 means the sum of some member, x, of the ring with that multiplicative identity. Since a ring is "closed under addition" x+ 1 is again a member of the ring and, since all the coefficients are members of the ring, all multiplications and addition in "p(x+ 1)" are defined in the ring.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Polynomials over a ring evaluated at a value?
  1. Polynomial ring (Replies: 4)

Loading...