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Can someone give me an idea how to find all $t \in \mathbb{R}$ such that $\mathbb{Q} [t]$ is isomorphic to $\mathbb{Q} [x] / (x^2+x-1)$ ?

(I only know that $\mathbb{Q} [\alpha]$ , where $\alpha$ is the equivalence class of $x$ in $\mathbb{Q} [x] / (x^2+x-1)$, is isomorphic to this same structure; but for plugging numbers in $\mathbb{Q} [\ ]$, I don't have any idea...)

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    First, can you see why $\mathbb{R}$ has a unique subfield isomorphic to $K = \mathbb{Q}[x]/(x^2 + x - 1)$? So $t$ must lie in this subfield. Second, any element of $K$ generates some subfield of $K$. What are the subfields of $K$?2012-01-09

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