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In a number field, the trace of an element over $\mathbb{Q}$ gives necessary conditions on the algebraic integers--the trace of an algebraic integer over $\mathbb{Q}$ is an integer.

But when the degree of the number field is large, computing the traces is laborious. My question is, does the trace over intermediate extensions give any information. For example if there an extension $\mathbb{Q}\subset K$ of degree 4, does the trace of an element in $K$ over $k$ give any useful information, where $k$ is any quadratic intermediate extension?

Spook
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The corresponding thing is true: if $\alpha$ in $K$ is an algebraic integer, then its trace down to any subfield $k$ is an integer of $k$.

Gerry Myerson
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    It's probably important to mention that this is true because we're in characteristic zero. Generally $\text{Tr}_{L/K}(\overline{\mathcal{O}_K})\subseteq\mathcal{O}_K$ is true when $L/K$ is separable. – Alex Youcis Feb 07 '13 at 22:27