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Thursday, January 29, 2015

**Abstract:** Recall that an abelian group is said to be indecomposable if it cannot be written as a direct sum of two non-trivial subgroups. Let F be a field and let F^* denote the multiplicative group of F. The problem we investigate is the following. When is F^* an indecomposable abelian group? We will solve this problem when the characteristic of F is not equal to 2. The answer involves Mersenne and Fermat primes, and one solution to this problem involves the famous Catalan's conjecture. We also have some partial results when the characteristic of the field is 2. I will discuss these results which are in joint work (arXiv:1407.3481) with Keir Lockridge.