Question in Proof of Fundamental theorem of Algebra using Galois Theory

Question

While self studying Galois theory from Thomas Hungerford. I have questions in the proof of Fundamental theorem of Algebra(Pg 266) .

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i have questions in 2 line of 2 nd paragraph of proof . Please note that ||.5.7 is Sylow 1 st Theorem. 1.Why H must have odd index? 2.Why fixed field E has odd dimension?