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It is classical that any such $\\mathsf{E}$ contains an element $\\beta\\in\\mathsf{E}$ that is normal over $\\mathbb F_q$, i.e., the conjugates $\\beta,\\beta^q,\\ldots,\\beta^{q^{n-1}}$ form an $\\mathbb F_q$-basis of $\\mathsf{E}$. 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