I know that:
$F(\alpha) = F[X]/f(x)$ where $F[x] = \{a_{n}x^{n} + a_{n-1}x^{n-1} + ... +a_{1}x+a_{0} \mid a_{i} \in F\}$ is ring of polynomials over F and $f(x)$ is irreducible polynomial in $F[X]$ and $\alpha$ is its root.
and $F(\alpha) = a_{d-1}\alpha^{d-1} + a_{n-1}\alpha^{d-2} + ... +a_{1}\alpha+a_{0} $
but I'm not getting the meaning of
$ Z[\alpha] = Z[X]/fZ[X]$ $ |$ $ f$ is irreducible in $\in$ $Z[X] $ of degree $d$ and $\alpha$ is root $\alpha = (X$ mod $fZ[X])$
$Z[\alpha] = Z.1 \oplus Z.\alpha \oplus Z.\alpha^{2} \oplus .... Z.\alpha^{d-1} $
why can't we write it as:
$Z(\alpha) = Z[X]/f(x)$ and
$Z(\alpha) = a_{d-1}\alpha^{d-1} + a_{n-1}\alpha^{d-2} + ... +a_{1}\alpha+a_{0} $
Secondly
$K$ and $Q$ are field and $K$ is an extension field of $Q$ of degree n. We write it as: $[K:Q]=n$.
but I'm not getting the meaning of:
$[\mathcal{O} : Z] =n$ are they same or different?
What is the difference between these ($[K:Q]=n$ , $[\mathcal{O} : Z] =n$) notations and when to use the specific one?.
Please try to explain it at your lowest possible level. Thank you.