TITLE: Correspondence between prime ideals and Galois orbits of affine points on an elliptic curve
QUESTION [5 upvotes]: In notes of prof. W.Stein -  http://wstein.org/edu/2010/581b/stein-algebraic_number_theory.pdf - the first paragraph of page 112 has the following told:
"When $K$ is a perfect field, the prime ideals correspond to the Galois orbits of affine points of $E(\bar{K})$."
I would be thankful if someone could elaborate what is meant here by the Galois orbits of points of $E$ and what is the implied correspondence.
Thank you!

REPLY [2 votes]: When $K = \bar{K}$, the correspondence defined by $P \mapsto \{r \in R: r(P) = 0\}$ is the well-known bijection (as follows from the Nullstellensatz, prime ideals correspond to irreducible algebraic sets), while when $K \neq \bar{K}$, any $\sigma(P)$ gets mapped to the same prime ideal when $\sigma$ runs through $Gal(\bar{K}/K)$.