Normal extension with simple, not cyclic Galois group Well, I can tell you "simple but not cyclic" means, if that's any help. A simple group is a group which has no nontrivial proper normal subgroup. If $p$ is prime then the (cyclic) group of order $p$ is simple, but there are (many) non-cyclic examples. The smallest is $A_5$, the alternating group on 5 symbols.
What is algebraic function theory? Let $F$ be a field. I believe that algebraic function field is to $F(t)$ what algebraic number field is to the rationals, that is, it's a finite extension.
Proof of Extended Euclidean Algorithm? The standard way to go at it is to prove that the smallest positive integer of the form $ax+by$ is the gcd of $a$ and $b$.
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