abcd...EFG - Ideal Behavior in Fields Relating to Iterated Polynomials

One branch of number theory is ramification theory, which studies the behavior of primes in L/K, where K is a number field with ring of integers O_K and L is an extension of K. More specifically, ramification theory asks: given a prime ideal p in O_K, what happens when p is lifted to O_L? In this project, we look at how the prime ideal \u3c2\u3e is factored in the ring of integers of the field L obtained by adjoining Q to the roots of the polynomial (x^2+c)^2+c for fixed c in N. Our polynomial of interest is the 2nd iterate of the polynomial $x^2+c$. We conjecture and attempt to prove: 1) For all c, the prime ideal \u3c2\u3e will ramify in L. 2) For all c congruent to 1 (mod 4), \u3c2\u3e totally ramifies in the ring of integers of L. 3) For c congruent to 2 (mod 4), \u3c2\u3e will either factor as p^4 or (p1)^4(p2)^4 for some p, p1, p2 in L. More specifically, for c congruent to \u3c2\u3e (mod 8), \u3c2\u3e will factor as p^4 and for c congruent to 6 (mod 8), \u3c2\u3e will factor as (p1)^4(p2)^4 in L. 4) For all c, \u3c2\u3e will not factor as p^2 for some fractional ideal p in L