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Is $(XY - 1)$ a maximal ideal in $k[[X]][Y]$, and if so, how can I see it?

It is at least prime because the generator is irreducible, and by the same argument it is maximal among all principal ideals. But I haven't gotten further than that - Finding units in the quotient ring didn't turn out well, either.

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    Is $k[[x]][y]/(xy-1) \cong k[[x]][x^{-1}]$ a field?2012-01-15
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    Thanks, this isomorphy is very helpful. I immediately thought that it is obvious that this is a field, but as it turns out, I'm not sure yet2012-01-15
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    Maybe this will help: what are the units in $k[[X]]$?2012-01-15
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    Ah yes, I vaguely remember having proven that all elements of $K[[X]]$ except for X are invertible, is that true? In that case, I'd be done.2012-01-15
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    @argon That's not quite true. The units are the formal power series having non-zero constant terms. So all non-zero elements look like $X^n \times (\text{a unit})$.2012-01-15
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    Of course, you're right! Thanks, I got it. The quotient is indeed a field, then.2012-01-15
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    Cool. Maybe I'll upgrade this to an answer later.2012-01-15
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    Bill and Dylan's comments are excellent. I have written an answer emphasizing the localization point of view. However I strongly encourage Dylan to upgrade his comment, more centered on the valuation aspect, to a complete answer.2012-01-15

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