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A quick question. What is the maximal ideal of the ring $\mathbb{R}[[x_1,\cdots,x_n]]$ of formal power series with coefficients in $\mathbb{R}$?

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Are you sure you should be using the word "the"? –  MartianInvader Feb 1 '13 at 23:30
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@MartianInvader, as the given ring is defined over a field then it is a local ring, so yes: the use of "the" is correct. –  DonAntonio Feb 1 '13 at 23:31
    
@martianInvader yes that's how it's written... –  Bruce Wayne Feb 2 '13 at 1:55

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up vote 4 down vote accepted

A quick answer: the set of elements with zero free coefficient.

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I'm sorry my knowladge of algebra is close to none, can you help me understand what you mean by free coefficient? –  Bruce Wayne Feb 2 '13 at 1:52
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What does an element in the ring looks like?: $$\sum_{(\nu)} a_{(\nu)}x_1^{\nu_1}x_2^{\nu_2}\cdot\ldots\cdot x_n^{\nu_n}=a_{(0,...,0)}+\ldots\;\;,\;a_{(\nu_1,...,\nu_n)}\in\Bbb R\,\,,\,\nu=(\nu_1,...,\nu_n)\in\left(\Bbb N\times \{0\}\right)^n $$ the free coefficient being $\,a_{(0,...,0)}\,$ . For example, in $\,7-3x_1+x_4^2-x_1^2x^2x_3^4+\ldots\,$ , the free coefficient is $\,7\,$ –  DonAntonio Feb 2 '13 at 10:15
    
Thanks so much you guys! –  Bruce Wayne Feb 2 '13 at 18:17

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