# 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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### Is the completion of $(k[x,y]/f)_\mathfrak{m}$ isomorphic to $k[[x]][y]/f$?

Let $k$ be an algebraically closed field and let $f\in k[x,y]$ be an irreducible polynomial with no constant term that is not a polynomial in $x$ alone. Is it the case that the completion of the ...
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### Is $\cos x$ irreducible as a power series?

Let $\mathbb{Q}_{\mathrm{ent}}[[x]]$ be the ring of entire functions with rational coefficients. Is $$\cos x \;=\; \sum_{n=0}^\infty (-1)^n\!\frac{x^{2n}}{(2n)!}$$ irreducible in ...
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### counterexample of formal power series over a commutative ring with identity [duplicate]

Let $A$ be a commutative ring with identity. Let $A[[x]]=\{\sum_{i=0}^{\infty}a_ix^i\mid a_i\in A\}$. Then it can be shown that if $f(x)\in A[[x]]$ is nilpotent, then $a_i$ are nilpotent for all $i$. ...
In the ring of formal power series $\Bbb R[[x]]$ it is easy to check by induction that $$1 = (1-x)(1 + x + x^2 + \cdots).$$ Does this derivation imply the same identity for those real or complex ...