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If anyone is interested, a thorough treatment is given by the book referenced in the comments here: http://math.stackexchange.com/questions/137495/artin-schreier-extensions-over-characteristic-two-fields

If anyone is interested, a thorough treatment is given by the book referenced in the comments here: https://math.stackexchange.com/questions/137495/artin-schreier-extensions-over-characteristic-two-fields

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Tait
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If anyone is interested, a thorough treatment is given by the book referenced in the comments here: http://math.stackexchange.com/questions/137495/artin-schreier-extensions-over-characteristic-two-fields