MHB Solving an Unsolved Math Problem: Ring A & Polynomials

[Fernando Revilla]

I quote an unsolved problem posted in another forum on December 5th, 2012.

Is there any ring A such that in A[x] some polynomial of degree 2 is equal to a polynomial of degree 4? Can you give me an example and explain how this could be true. Thanks in advance!

 

[Fernando Revilla]

It is not possible. According to the formal definition of polynomial, if $p$ has degrre $2$ and $q$ degrre 4 then

$$p=(a_0,a_1,a_2,0,\ldots,0,\ldots)\qquad (a_2\neq 0)\\ q=(b_0,b_1,b_2,b_3,b_4,0,\ldots,0,\ldots)\quad (b_4\neq 0)$$

so, $p\neq q$. Another thing is that if $p$ and $q$ can determine the same polynomical function, and the answer is affirmative. For example, choose in $\mathbb{Z}_2[x]$ the polynomials $p(x)=x^2$ and $q(x)=x^4$.