  This paper studies the free group of rank two from the point of view of
Stallings core graphs. The first half of the paper examines primitive elements
in this group, giving new and self-contained proofs for various known results
about them. In particular, this includes the classification of bases of this
group. The second half of the paper is devoted to constructing a counterexample
to a conjecture by Miasnikov, Ventura and Weil, which seeks to characterize
algebraic extensions in free groups in terms of Stallings graphs.
