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In answer to the second part of your question, there is a paper by Pleasants [Zbl 0328.12008][Zbl 0328.12008],``The number of generators of the integers a number field''``The number of generators of the integers a number field'' in which it is shown that the number of generators is at most $\lceil log_2(n) \rceil$$\lceil \log_2(n) \rceil$, with equality when $2$ splits completely in the field.

In answer to the second part of your question, there is a paper by Pleasants [Zbl 0328.12008],``The number of generators of the integers a number field'' in which it is shown that the number of generators is at most $\lceil log_2(n) \rceil$, with equality when $2$ splits completely in the field.

In answer to the second part of your question, there is a paper by Pleasants [Zbl 0328.12008],``The number of generators of the integers a number field'' in which it is shown that the number of generators is at most $\lceil \log_2(n) \rceil$, with equality when $2$ splits completely in the field.

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In answer to the second part of your question, there is a paper by Pleasants [Zbl 0328.12008],``The number of generators of the integers a number field'' in which it is shown that the number of generators is at most $\lceil log_2(n) \rceil$, with equality when $2$ splits completely in the field.