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    $[\QQ(\zeta+\zeta^{-1}):\QQ]=\varphi(n)/2$ for a primitive $n$th root of unity $\zeta$

    P-XT3LV

    Area
    Algebra
    Topics
    Field Extensions, Galois Theory, Roots of Unity
    Status
    draft

    Let \(n>2\) and let \(\zeta_n\) be a primitive \(n\)th root of unity. Prove \[ [\QQ(\zeta_n+\zeta_n^{-1}):\QQ] =\frac{\varphi(n)}2. \]

    Solution

    Set \[ K=\QQ(\zeta_n+\zeta_n^{-1}). \] The element \(\zeta_n\) satisfies \[ x^2-(\zeta_n+\zeta_n^{-1})x+1=0, \] so \[ [\QQ(\zeta_n):K]\le2. \]

    Because \(n>2\), the primitive root \(\zeta_n\) is nonreal. On the other hand, \[ \zeta_n+\zeta_n^{-1}=2\cos(2\pi/n)\in\RR, \] so \[ K\subseteq\RR. \] Therefore \(\zeta_n\notin K\), and the extension is proper. Hence \[ [\QQ(\zeta_n):K]=2. \]

    The cyclotomic degree is \[ [\QQ(\zeta_n):\QQ]=\varphi(n). \] By the tower law, \[ \varphi(n) =[\QQ(\zeta_n):K][K:\QQ] =2[K:\QQ]. \] Therefore \[ [K:\QQ]=\frac{\varphi(n)}2. \]

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