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Abstarct-Algebra

In Abstarct-Algebra , some of the proofs of polynomial rings are below.

Polynomial Rings

Definition of Polynomial Ring :

  Let R be a ring . An infinite sequence \( (a_{0}, a_{1}, a_{2},....a_{n}, ......)\) of elements of R, with at most a finite number of non-zero terms, is called a Polynomial over the ring R.

Q) If \(f(x),g(x)\epsilon F(x)\) and g(x)|f(x) then prove that \((f(x))\subset (g(x))\) .

Proof:

         g(x)|f(x) \(\Rightarrow \) f(x)=a(x)g(x), for some a(x)\(\epsilon  \)F(x) .

          Let h(x)\(\epsilon\)((f(x)) then h(x)=b(x)f(x) , where b(x) is a                                

           polynomial function in F(x).

          h(x)=b(x)f(x)

                  =b(x)a(x)g(x)

                   =c(x)g(x)   , where c(x)=b(x)a(x)\(\epsilon\)F(x).

           h(x) \(\epsilon\) (g(x))

           If  h(x)  \(\epsilon\)  (f(x)) \(\Rightarrow\)   h(x)\(\epsilon\)(g(x))

           Hence (f(x))\(\subset\)(g(x)).

Q2)   If \(f(x),g(x)\epsilon F(x)\) and f(x), g(x) are co-primes and f(x)|h(x),          g(x)|h(x) then prove that f(x)g(x)|h(x).

Proof :   f(x), g(x) are co-primes means no common factor other than 1.

             f(X)|h(x) \(\Rightarrow\) h(x)=a(x)f(x)  where a(x) \(\epsilon\)F(x).

             If  g(x)|a(x)f(x) \(\Rightarrow\) g(x)|a(x) as g(x) does not divide f(x).

             g(x)|a(x)  \(\Rightarrow\) a(x)=b(x)g(x) , b(x) \(\epsilon\)F(x).

             but  we have h(x) = a(x)f(x)

                                  h(x) = b(x)g(x)f(x)

                Therefore f(x)g(x)|h(x) .

            Hence it is proved.

What we learned in the branch of Abstarct-Algebra , Polynomial Rings:

  • If \(f(x),g(x)\epsilon F(x)\) and g(x)|f(x) then  \((f(x))\subset (g(x))\)
  • If \(f(x),g(x)\epsilon F(x)\) and f(x), g(x) are co-primes and f(x)|h(x),          g(x)|h(x) then  f(x)g(x)|h(x).
  • Two functions are said to be Co-Primes if they have only 1 as their common factors, i.e., no common zeros .


In the following page it has the definition of Irreducible Polynomial and its examples .


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