# Tagged Questions

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### Why does the only maximal of $k[[X_1,\ldots,X_n]]$ is $(X_1,\ldots,X_n)$?

I'm trying to understand in this book why the only maximal of $k[[X_1,\ldots,X_n]]$ ($k$ field) is $(X_1,\ldots,X_n)$: If I prove $rad(k[[X_1,\ldots,X_n]])\subset (X_1,\ldots,X_n)$, (where $rad$ is ...
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### Unit of power series ring

Is there any way to calculate the multiplicative group of the units of power series ring $k[[x]]$ where $k$ is a field
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### Power series expression of x in terms of y

Let $R$ be a commutative ring with identity and $y=a_1x+a_2x^2+a_3x^3+.....$ be a power series in $R[[x]]$ such that $a_1$ is an unit in $R$. Does there exists a way to express x as a power series ...
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### Necessary and sufficient conditions for a polynomial in $\mathbb{Z}[t]$ to have an $n$th root in $\mathbb{Z}[[t]]$

Let $p(t) = \sum p_k t^k$ be a polynomial in $\mathbb{Z}[t]$, with $p_0=1$. Is there a necessary and sufficient condition (congruence or other) on the coefficients $p_k$ such that $p(t)$ admits a ...
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### Enhancing the monoid structure over a finite alphabet to prove Arden's rule

Suppose you have a finite-state, deterministic automaton, that you wish to convert to a regular expression. A common method, perhaps easier to apply by hand that Yamada's algorithm, is to reduce the ...
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### Embed $K(x)$ into $K[[x]]$

How to see formally/algebraically that the field of rational functions $K(x)$ embeds into the ring of formal power series $K[[x]]$?
Say if I define a power series over some arbitrary field $F$ as $$a = \sum^{ \infty }_{i = 0} a_{i} X^{i}$$ Then can I say: ab = \sum^{ \infty }_{i = 0} \sum^{ \infty }_{j = 0} a_{i} b_{j} X^{i ...