0
$\begingroup$

List all the elements of $\frac{\mathbb Z_{2}[x]}{}$ (the set of remainders)

Please verify my understanding:

  • Since the polynomial is of degree 3, the remainders must have degree at most 2.
  • We are working in $\mathbb Z_{2}[x]$ so the coefficients must be 0 or 1

Is it $\{0, 1, x, x^2, x+1, x^2+1\}$ ?

  • 0
    Now that you know what the seven nonzero elements of your field are, you should improve your understanding of its structure by writing down the powers of (for instance) $x$ in order, and noticing that you now have a logarithm table for the multiplicative structure of the field. I always do this when I'm working with a field that has more than $4$ elements.2012-11-08

3 Answers 3

1

The elements in $\mathbb Z[x]/$ are equivalence classes represented by polynomials of degree less than $3$. So, formally speaking, none of the elements you list are in $\mathbb Z[x]/$. However, these elements do represent elements in $\mathbb Z[x]/$. The elements you list all represent different elements in $\mathbb Z[x]/$ but not all of them. You are missing $2$ more polynomials: $x^2+x$ and $x^2+x+1$. This will give you a complete list of representatives.

In general, given a polynomial $p(x)\in F[X]$, where $F$ is a field, the ring $R=F[x]/$ has cardinality $|F|^d$ where $d$ is the degree of $p(x)$. Every element in $R$ is an equivalence class of the form $q(x)+$, and a complete list of representatives is given by all polynomials in $F[x]$ of degree smaller than $d$.

2

$\{0, 1, x, x^2, x+1, x^2+1,x^2+x,x^2+x+1\}+$

  • 0
    ah of course, missing $x^2+x$ and $x^2+x+1$! thanks2012-11-08
2

Hint $ $ Polynomials in $\Bbb Z_2[x]$ of degree $\le 2$ have form $\rm\ \Bbb c_0\! + c_1 x +\, c_2 x^2\:$ for $\rm\:c_i \in \Bbb Z_2,\:$ so there are $\rm\: 2\cdot 2\cdot 2 = 8\:$ in total, not 6. Indeed, your proposed coset rep's are not closed under addition, so they cannot comprise a complete set of cosets rep's for the quotient ring.