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Using the recently uncovered connection between Hopf Galois structures and skew left braces, we introduce a method to quantify the failure of surjectivity of the Galois correspondence from subHopf algebras of $H$ to intermediate subfields of $L/K$, given by the Fundamental Theorem of Hopf Galois Theory. Suppose $L \\otimes_K H = LN$ where $N \\cong (G, \\star)$. Then there exists a skew left brace $(G, \\star, \\circ)$ where $(G, \\circ) \\cong \\Gamma$. 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Childs","submitted_at":"2018-02-09T21:11:13Z","abstract_excerpt":"Let $L/K$ be a Galois extension of fields with Galois group $\\Gamma$, and suppose $L/K$ is also an $H$-Hopf Galois extension. Using the recently uncovered connection between Hopf Galois structures and skew left braces, we introduce a method to quantify the failure of surjectivity of the Galois correspondence from subHopf algebras of $H$ to intermediate subfields of $L/K$, given by the Fundamental Theorem of Hopf Galois Theory. Suppose $L \\otimes_K H = LN$ where $N \\cong (G, \\star)$. Then there exists a skew left brace $(G, \\star, \\circ)$ where $(G, \\circ) \\cong \\Gamma$. 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