Ranks of elliptic curves over cyclic cubic, quartic, and sextic extensions

Let E be an elliptic curve defined over a number field K. I prove the following theorems:;Theorem 1 ([FKK12]). If K is a number field containing the cube roots of unity, and E is any elliptic curve defined over K, then there are infinitely many Z/3Z-extensions of K over which E rises in rank. Theorem 2. Let E be an elliptic curve defined over Q with Z/2Z x Z/6Z torsion. Then there are infinitely many cyclic cubic extensions of Q over which E increases in rank.;Theorem 3. If K is a number field containing i and if E is an elliptic curve over K, then there are infinitely many Z/4Z-extensions of K over which E gains in rank.;Theorem 4. Let K be a number field containing the cube roots of unity and let E be any elliptic curve defined over K. Then there are innitely many Z/6Z-extensions of K over which E increases in rank. The first theorem is a result of Fearnley, Kisilevsky, and Kuwata [FKK12] and the second theorem is a special case of another result in the same paper. I give new proofs of these results.