Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

EDIT: The first paragraph has been indicated as inaccurate, please see this answer.

When we have something like $\mathbb{Z}_2[x]/(x^2 + x + 1)$, we understand that this means the set of polynomials over indeterminate $x$ where the coefficients are drawn from $\mathbb{Z}_2 = \{0,1\}$ with degree less than $\text{deg} (x^2 + x + 1)$.

So, in this set we can include polynomials like $x$, $x+1$ and even $1$, but not $x^2+1$ or $2x+1$. This is okay so far with me.

But in some lecture note (Rings and fields, Sergei Silvestrov, Spring term 2011, Lecture 5; page 3, Example 1 and subsequent ones) I find representations like, $\mathbb{Z}_2[x]/(x^2 + x + 1) = \{[0], [1], [x], [x + 1]\}$.

I fail to understand why the square brackets have been put around the polynomials.

Answer to this question says that this depends on the context and may be same as parentheses or may mean braces to represent sets.

Your help will be appreciated.

share|cite|improve this question
It's an equivalence class. All polynomials in $[0]$ are equivalent to the polynomial $0$ up to polynomial multiples of $x^2+x+1$. The nice thing is that the set of resulting equivalence classes inherits the structure of the original space. Hence the name quotient space. – Raskolnikov May 30 '13 at 10:07
@Raskolnikov Thanks for pointing to the right direction. Discussions in the book by Hungerford cleared things substantially. – Masroor May 30 '13 at 19:49
up vote 2 down vote accepted

Your first three lines aren't accurate: that symbol is for a field , represented by that quotient ring, the elements of which can be expressed as polynomials of degree zero or degree one in one of the roots of the polynomial $\;x^2+x+1\;$ over $\,\Bbb F_2\,$.

Thus, the elements in that "set" are not polynomials but expressions of the form $\,w, w+1,$ , etc., where $\, w^2+w+1=0\pmod 2\,$ , and precisely in order to avoid this confusion is that Silvestrov uses the symbols he does...

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.