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Some standard examples on various quals seems to be computing units/class numbers etc. of the ring $\mathbb{Q}(\alpha)$, where $\alpha$ is a root of either $X^3+aX+b$ or $X^5+aX+b$.

My questions is the following: What are some standard tricks that can be used to deduce that a particular unit is actually a fundamental unit?

I'm familiar with class field theory and L-series, so I don't mind higher-level methods. I'm sort of trying to figure out what's usable during an exam. Books on number theoretic algorithms only cover pretty general cases and these algorithms are too computationally intensive to be done manually.

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Show that no other unit can have smaller norm in some real or complex embedding. This is feasible in the quadratic case; it might be feasible in the cubic case. – Qiaochu Yuan Mar 16 '11 at 21:14
Well, the quadratic case is trivial by just looking at the corresponding a Pell equation. This doesn't seem to work as easily for higher-degree roots. – user8351 Mar 16 '11 at 22:11
Yes, but that doesn't contradict what I said. – Qiaochu Yuan Mar 16 '11 at 22:40

For applications to diophantine equations it is usually sufficient to know that a unit is not a square or a cube; this can be verified quickly by showing that the unit in question is not a square or a cube in a residue class ring modulo a suitable chosen prime ideal. In particular, squares must be totally positive.

For finding an upper bound for the exponent n such that your unit is an n-th power modulo roots of unity, you will have to get your hands dirty by applying to geometric methods (Minkowski etc.). The only clean example I know beyond the quadratic case is that of cubic fields discussed by Artin in his book on algebraic numbers (see this question).

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