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.
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.