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Some confusion of the concepts about polynomials

  1. Nov 10, 2008 #1
    I'm not reading a text in English, so I should clarify some notation first:
    P[X], is the set of all polynomials.
    P[[X]], is the set of formal power series, which may infinitely many nonzero coefficients.
    I'm asked to prove that X is the only prime(?correct term?) element in P[[X]]. By saying 'only', I mean uX and X is considered to be the same, if u is the unit in P.( of course every nonzero element in P is the unit)

    The prime element f in P[X] is defined to be so that there's no other polynomial g with 0<deg(g)<deg(f) such that g|f. And I know that every f with deg(f)=1 is a prime polynomial.
    But deg(f) makes no sense in P[[X]]. Then what kind of element is the prime one in P[[X]]?My book does not tell me, but introduce w(f) to me, which is defined to be the index of the first nonzero coefficient. More precisely, if f=a_n*X^n+a_(n+1)*X^(n+1)+.....then w(f)=n


    for instance, how to show that 1+X is not prime?
     
  2. jcsd
  3. Nov 10, 2008 #2

    Hurkyl

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    What you describe sounds like the notion of an irreducible element. (But in these rings, every irreducible is prime, and vice versa) a is irreducible iff:
    1. a is not a unit
    2. If b divides a, then b is a unit or b = ua for some unit u

    Note that for P[X], this is equivalent to the definition you gave.
     
  4. Nov 10, 2008 #3
    thanks!

    let f=a_0+a_1*X+.....
    so if a_0!=0,that is w(f)=0 then there is a g such that fg=1, which implies that f is a unit.(g can be found by comparing the coefficients of fg and 1)

    Suppose X is not irreducible, then X=gh, where w(g)>=1 and w(h)>=1 (which means g and h are not units). Hence 1=w(X)=w(g)+w(h)>=2 and we get a contradiction.

    if w(f)=s>0. then f=X^(s)*h, where h has the form h=b_0+b_1*X+....and w(h)=0.
    It is sufficient to show that X^s is not irreducible, but this is obviously true if s>=2. Hence X is the only irreducible element in P[[X]].

    It works, right?
     
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