We prove a new lower bound on the field size of locally repairable codes (LRCs). Additionally, we construct maximally recoverable (MR) codes which are cyclic. While a known construction for MR codes has the same parameters, it produces non-cyclic codes. Furthermore, we prove both necessary conditions and sufficient conditions that specify when the known non-cyclic MR codes may be permuted to become cyclic, thus proving our construction produces cyclic MR codes with new parameters. Furthermore, using our new bound on the field size, we show that the new cyclic MR codes have optimal field size in certain cases. Other known LRCs are also shown to have optimal field size in certain cases.