How does one tell if a number field contains a certain element?

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I think your question is not well-defined. If you had a number field $K$ along with an embedding $K \to \mathbb{C}$ then the question makes sense. When you construct a number field in Sage I believe it is an abstract field since there may be many different embeddings of $K$ into $\mathbb{C}$.

Now, if your field $K$ is the splitting field of an irreducible polynomial $f$ over $\mathbb{Q}$ then you could test if $f(\sqrt[4]{29})$ is zero or not.