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 ...
2
votes
2answers
183 views

Evaluation morphisms of formal power series and nilpotent elements

Given a commutative ring $A$, and a finitely presented (associative) $A$-algebra $B$, show that a morphism of $A$-algebras $A[[x]] \longrightarrow B$ is given by evaluation at an nilpotent element $ ...
0
votes
1answer
56 views

Convergence of formal power series substitution

Prove that the substitution of formal power series $F(G(x))=\sum_{k\geq0}f_k \frac{G(x)^k}{n!}$ converges for every $F$ if and only if $G(0)=0$
1
vote
1answer
74 views

Question about the infinite products of formal power series

I need a proof for this: Let $(F_j)_{j\ge 0}$ be a sequence of formal power series. The infinite product $\prod_{j\geq0}(1+F_j(x))$, where $F_j(0)=0$, converges if and only if ...
3
votes
2answers
338 views

Formal power series, the Chain Rule and the Product Rule.

Definitons Let $$\mathbb{C}[[x]] := \left\{ \sum_{n\geq 0} a_n x^n : a_n \in \mathbb{C} \right\}$$ be the set of formal power series of $x$. Exercise i) If $F_1(x)$ and $F_2(x)$ are power series ...
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
1answer
46 views

Kernel of the evaluation map on a power series ring

Let $R$ be a commutative ring with unity and $r \in R$ a nilpotent element. Is it true that if $f \in R[[\epsilon]]$ satisfies $f(r) = 0$, then $(\epsilon - r) | f$ in $R[[\epsilon]]$? I tried solving ...
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 ...
-1
votes
5answers
147 views

Expanding the power series

$$g_2(\epsilon^1 \phi_1+ \epsilon^2 \phi_2+ \epsilon^3 \phi_3+\cdots)^2+ g_3(\epsilon^1 \phi_1+ \epsilon^2 \phi_2+ \epsilon^3 \phi_3+\cdots)^3+g_4(\epsilon^1 \phi_1+ \epsilon^2 \phi_2+ \epsilon^3 ...
3
votes
2answers
274 views

Logarithm Series: Symbol Manipulation Proof that $\log(x) + \log(y) = \log (xy)$

Let $R$ be a ring with 1. Define a formal power series $$\log(x)=\sum_{m=1}^\infty (-1)^{m+1}\frac{(x-1)^m}{m}.$$ I would like to show using only manipulations of the power series (pretending we know ...
0
votes
2answers
123 views

Factoring polynomials over a power series ring

Could anyone tell me why $g(x, y) = x^3 - y^2$ is irreducible in $\mathbb C[[x]][y]$ while $f(x, y) = x^3 - x^2 + y^2$ is not?
1
vote
2answers
139 views

Does $R[[x]]$ have a basis and is it countable?

We know that $R[x]$ is not finitely generated as an $R$-module and has a basis of $\{1,x,x^2,\ldots\}$. I started thinking about whether or not $R[[x]]$ has a basis, and if it does have a basis, if it ...
3
votes
1answer
149 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
163 views

Showing that $R(x)$ is a proper subset of $R((x))$ if R is a field

I would like to show that if $R$ is a field, then $R(x)$ is a proper subset of $R((x))$, where $R(x)$ is the ring of rational functions, and $R((x))$ is the ring of formal Laurent series. If $f \in ...
3
votes
3answers
624 views

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
1
vote
0answers
79 views

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 ...
14
votes
1answer
231 views

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 ...
4
votes
3answers
449 views

Ring of formal power series

Let $$\mathbb{C}[[x]] := \left\{ \sum_{n\geq 0} a_n x^n \;\colon\; a_n \in \mathbb{C} \right\}$$ be the set of formal power series of $x$ and $$F(x) = \sum_{n\geq 0} a_n x^n \in \mathbb{C}[[x]], \; ...
4
votes
2answers
494 views

Proofs for formal power series

Definitons Let $$\mathbb{C}[[x]] := \left\{ \sum_{n\geq 0} a_n x^n : a_n \in \mathbb{C} \right\}$$ be the set of formal power series of x and $$F(x) = \sum_{n\geq 0} a_n x^n, \; G(x) = \sum_{n\geq 0} ...
4
votes
1answer
76 views

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 ...
2
votes
3answers
189 views

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 $K[[x]]$?
11
votes
3answers
1k views

Product of two power series

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 ...
8
votes
5answers
441 views

Formal power series - a question

I've been reading generationgfunctionology by Herbert S. Wilf (you can find a copy of the second edition on the author's page here). On page 33 he does something I find weird. He wants to shuffle the ...