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Let $A$ be a local ring with maximal ideal $m$ that is $m$-adically complete, and assume $1/2 \in A^\times$. I've read in several places that for any $x \in m$, a square root of $1 + x$ in $A$ is given by the binomial series

$$ \sum_{n = 0}^\infty {{1/2} \choose n} x^n, $$ where ${{1/2} \choose n} = \frac{(1/2)(1/2 - 1) \cdots (1/2 - n + 1)}{n!}$. I don't understand why these binomial coefficients make sense in $A$. We have $n!$ in the denominator, so why is $n! \in A^\times$?

Jackie
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    A ratio being in $A$ does not require that the denominator has to be a unit. For example, 15/5 is an integer even though 5 is not a unit in the integers. Or, to give an example with a local ring, 15/5 is in ${\mathbf Z}_5$ (the 5-adic integers) even though 5 is not a unit in ${\mathbf Z}_5$. The point is that the rational number $\binom{1/2}{n}$ in reduced form has a denominator that is a power of 2, so it makes sense in $A$. See http://math.stackexchange.com/questions/136206/show-that-sqrt1t-lies-in-mathbbz1-2-t. – KCd Apr 29 '12 at 19:12
  • Ah yes, I see. Thanks! – Jackie Apr 29 '12 at 19:19
  • @KCd Please consider converting your comment into an answer, so that this question gets removed from the unanswered tab. If you do so, it is helpful to post it to this chat room to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see here, here or here. – Julian Kuelshammer Jun 25 '13 at 20:10
  • Julian: You could post my comment as an answer if you wish. – KCd Jun 25 '13 at 22:03

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As KCd remarks:

A ratio being in $A$ does not require that the denominator has to be a unit. For example, $15/5$ is an integer even though $5$ is not a unit in the integers. Or, to give an example with a local ring, $15/5$ is in $\Bbb Z_5$ (the $5$-adic integers) even though $5$ is not a unit in $\Bbb Z_5$. The point is that the rational number $1/2 \choose n$ in reduced form has a denominator that is a power of $2$, so it makes sense in $A$. See this question for details.

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Lord_Farin
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