Our lecturer defined the following:
Let $K=\mathbb Q(\sqrt d)$ be a quadratic field and $p$ a prime number, then
(1) $\ p$ is ramified in $K$ if $\mathcal O_K⁄(p)\cong \mathbb F_p [x]⁄(x^2)$
(2) $\ p$ is split in $K\ \ \ \ \ \ $ if $\mathcal O_K⁄(p)\cong \mathbb F_p[x]^2$
(3) $\ p$ is inert in $K\ \ \ \ \ \ $ if $\mathcal O_K⁄(p)\cong \mathbb F_{p^2}$
And he stated:
Let $K$ and $p$ be as above. Then:
$p$ is ramified $\iff p$ divides the discriminant disc$(K)$ of $K$
$p$ split $\iff p\nmid \text{disc(K)}$ and d is a QR mod p
$p$ is inert $\iff p\nmid \text{disc(K)}$ and d is not a QR mod p
To be honest, I have a little bit of trouble wrapping my head around these definitions. Also, I don't seem to be able to find any useful sources online, which provide examples or (not to mention) use similar definitions. All I can find are things like "totally split", "remains prime in", etc. and a lot of different notation.
Hence my question, are these definitions actually common and is there any way of explaining them in a more approachable manner?
edit:
Just in case somebody is interested, the following texts, which I found after further research, appeared quite useful to me:
An Introduction to Number Theory, by G. Everest:
http://www.amazon.co.uk/dp/1852339179
Specifically Chapter 4.4 The Ideal Class Group and Section 4.4.1 Prime IdealsThe lecture notes of Robin Chapman:
http://empslocal.ex.ac.uk/people/staff/rjchapma/notes/ant.pdf (p. 16-18) http://empslocal.ex.ac.uk/people/staff/rjchapma/notes/ant2.pdf (p. 38-39 & Chapter 5)