1
vote
1answer
41 views

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 ...
1
vote
1answer
77 views

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 ...
5
votes
1answer
56 views

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 ...
0
votes
0answers
23 views

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 $i$. ...
5
votes
0answers
103 views

“Evaluation Homomorphisms” for Formal Power Series

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 ...
1
vote
1answer
43 views

Trying to show that the product of two power series equals 1.

I've reduced a large homework problem to the following smaller problem. Let $P = \sum_{i=0}^\infty a_i X^i$ denote a formal power series over a field. Assume $a_0 \neq 0$, and define $Q = ...
2
votes
3answers
70 views

Is the degree function well behaved over power series?

For non-zero formal polynomials $x$ and $y$ it holds that $\deg(xy)=\deg x + \deg y$. Allowing for infinite degrees, does this formula hold for arbitrary non-zero power series? And is there a ...
1
vote
1answer
48 views

Notation for the coefficient of the $i$th term of formal power series.

What notation is standard for the coefficient of $X^i$ in a formal power series $P$? I was thinking of $X^i \cdot P$, by analogy with the dot product.
5
votes
1answer
77 views

An identity involving the powers of a nilpotent element in a unital commutative ring

Suppose $R$ is a commutative unital ring with identity $1$ such that the equation $nx = 1$ has a unique solution for each integer $n \ge 1$, and let $\xi$ be a nilpotent element of $R$ with nilpotency ...
3
votes
1answer
172 views

How does one show that this tensor product is not torsion-free?

I am having trouble showing that a particular tensor product is not torsion-free. Let $ R = k[[x,y]] $, where $ k $ is a field (this is the ring of formal power series in $ x $ and $ y $ with ...
3
votes
1answer
150 views

Inverses in the ring of formal power series over a field.

Let $\mathbb{F}$ be a field, and consider $\mathbb{F}[[x]]$, the ring of formal power series with coefficients in $\mathbb{F}$, i.e. the set of expressions of the form $$\sum_{n=0}^{\infty}a_n ...
2
votes
3answers
85 views

If $a_0\in R$ is a unit, then $\sum_{k=0}^{\infty}a_k x^k$ is a unit in $R[[x]]$

Let $R$ a ring, and let $$\displaystyle R[[x]]=\left\{\sum_{k=0}^{\infty}a_k x^k\;\middle\vert\; a_k\in R\right\}$$ with addition and multiplication as defined for polynomials. We have that $R[[x]]$ ...
5
votes
3answers
141 views

Is this action of $\mathbb F[[x]]$ on $\bigoplus_{i=0}^{\infty}\mathbb F$ natural?

The title of my question has a field $\mathbb F$ in it, but to make sure I'm not losing anything, I would like to introduce my question in full generality. But still, I will be happy with an answer in ...