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I am starting Galois theory in my class and have run into a small problem in regards to quotient rings. Let me illustrate this confusion with an example so I can also illustrate what I understand to make it clear.

Let $F$ be a field, $F[x]$ a polynimial ring, the domain of all polynomials $f(x)$ with coeffients in $F$. Let $(I(x))$ be the principal ideal generated by polynomial $I(x)$ in $F[x]$. Then we can construct a quotient ring $F[x]/(I(x))$.

Now the elements in $F[x]/(I(x))$ are the residue equivalence classes created by the equivalence relation "$\equiv mod \; I(x)$". So $f(x) \equiv g(x)\; mod\; I(x) \;\iff I(x) \;|\; g(x)-f(x)$. By division algorithm, we know any $f(x) \in F[x]/(I(x))$ can be expressed as $f(x) = q(x)I(x) + r(x)$.

Using this information above, lets say $I(x) = x^2$ and let me get to the point.

Then,

$F[x]/(x^2) =\{\alpha + \beta x \;| \;\alpha,\beta \in F \; and \;x^2 = 0\}$

OR

$F[x]/(x^2) =\{\alpha + \beta x + (x^2)\;| \;\alpha,\beta \in F \; and \;x^2 = 0\}$?

I know that elements are the remainders after division and $x^2 =0$ in our set, but I keep seeing elements expressed both ways. Which one is correct?

I am under the impression people use the first notation when just directly speaking of the cosets of $I(x)$ instead of explicitly writing "$+\; I(x)$" each time. Thank you.

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  • $\begingroup$ It's just abuse of notation. Technically, it's $\alpha+\beta x+(x^2)$, but sometimes if we want to be lazy we'll write $\overline{\alpha+\beta x}$. Then, if we want to be even more lazy, we just write $\alpha+\beta x$. Alternatively, you can think of it the following way: the correct notation is the second you've described, but this ring is actually isomorphic to the ring $\{\alpha+\beta x\mid\alpha,\beta\in F,x^2=0\}$ $\endgroup$ – Alex Mathers Nov 4 '16 at 0:34
  • $\begingroup$ @Alex Mathers Thanks for clearing that up. So when someone is mentioning the element $[x]$ they also mean $\bar x$? $\endgroup$ – Mir Nov 4 '16 at 0:44
  • $\begingroup$ yes, that's correct. $\endgroup$ – Alex Mathers Nov 4 '16 at 9:12
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It is exactly the same as you represent cosets in the quotient ring of integers modulo an ideal (or subgroup).

You can write elements of the elements of quotients as $0+5\mathbf{Z}, 1+5\mathbf{Z}, 2+5\mathbf{Z},3+5\mathbf{Z},4+5\mathbf{Z}$ or simply as $\bar 0, \bar 1, \bar 2, \bar 3, \bar 4, \bar 4, \bar5=\bar0,$. In the first case the cosets are shown as the arithmetic progressions (a subset of the ring); in the second case elements of cosets representing each coset is listed.

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