Restricted power series

<h2 id="definition">Definition</h2>
Let A be a <a href="/facts/Linearly_topologized_ring/5w5TpKx9">linearly topologized ring</a>, separated and complete and 
 
 
 
 {
 
 I
 
 λ
 
 
 }
 
 
 {\displaystyle \{I_{\lambda }\}}
 
 the fundamental system of open ideals. Then the ring of restricted power series is defined as the <a href="/facts/Projective_limit/f6QecRUu">projective limit</a> of the polynomial rings over 
 
 
 
 A
 
 /
 
 
 I
 
 λ
 
 
 
 
 {\displaystyle A/I_{\lambda }}
 
:

A
 ⟨
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 ⟩
 =
 
 
 
 lim
 ←
 
 
 
 λ
 
 
 ⁡
 A
 
 /
 
 
 I
 
 λ
 
 
 [
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 ]
 
 
 {\displaystyle A\langle x_{1},\dots ,x_{n}\rangle =\varprojlim _{\lambda }A/I_{\lambda }[x_{1},\dots ,x_{n}]}
 
.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
In other words, it is the <a href="/facts/Completion_(ring_theory)/Q0vjA9JI">completion</a> of the polynomial ring 
 
 
 
 A
 [
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 ]
 
 
 {\displaystyle A[x_{1},\dots ,x_{n}]}
 
 with respect to the filtration 
 
 
 
 {
 
 I
 
 λ
 
 
 [
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 ]
 }
 
 
 {\displaystyle \{I_{\lambda }[x_{1},\dots ,x_{n}]\}}
 
. Sometimes this ring of restricted power series is also denoted by 
 
 
 
 A
 {
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 }
 
 
 {\displaystyle A\{x_{1},\dots ,x_{n}\}}
 
.
Clearly, the ring 
 
 
 
 A
 ⟨
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 ⟩
 
 
 {\displaystyle A\langle x_{1},\dots ,x_{n}\rangle }
 
 can be identified with the subring of the formal power series ring 
 
 
 
 A
 [
 [
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 ]
 ]
 
 
 {\displaystyle A[[x_{1},\dots ,x_{n}]]}
 
 that consists of series 
 
 
 
 ∑
 
 c
 
 α
 
 
 
 x
 
 α
 
 
 
 
 {\displaystyle \sum c_{\alpha }x^{\alpha }}
 
 with coefficients 
 
 
 
 
 c
 
 α
 
 
 →
 0
 
 
 {\displaystyle c_{\alpha }\to 0}
 
; i.e., each 
 
 
 
 
 I
 
 λ
 
 
 
 
 {\displaystyle I_{\lambda }}
 
 contains all but finitely many coefficients 
 
 
 
 
 c
 
 α
 
 
 
 
 {\displaystyle c_{\alpha }}
 
.
Also, the ring satisfies (and in fact is characterized by) the <a href="/facts/Universal_property/5XJBYMNz">universal property</a>:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> for (1) each <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous</a> <a href="/facts/Ring_homomorphism/UptcMpFk">ring homomorphism</a> 
 
 
 
 A
 →
 B
 
 
 {\displaystyle A\to B}
 
 to a linearly topologized ring 
 
 
 
 B
 
 
 {\displaystyle B}
 
, separated and complete and (2) each elements 
 
 
 
 
 b
 
 1
 
 
 ,
 …
 ,
 
 b
 
 n
 
 
 
 
 {\displaystyle b_{1},\dots ,b_{n}}
 
 in 
 
 
 
 B
 
 
 {\displaystyle B}
 
, there exists a unique continuous ring homomorphism

A
        ⟨
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        ⟩
        →
        B
        ,
        
        
          x
          
            i
          
        
        ↦
        
          b
          
            i
          
        
      
    
    {\displaystyle A\langle x_{1},\dots ,x_{n}\rangle \to B,\,x_{i}\mapsto b_{i}}

extending 
 
 
 
 A
 →
 B
 
 
 {\displaystyle A\to B}
 
.

<h2 id="tate-algebra">Tate algebra</h2>
In <a href="/facts/Rigid_analysis/2p3fUBHc">rigid analysis</a>, when the base ring A is the <a href="/facts/Valuation_ring/GpLuHXS0">valuation ring</a> of a complete non-archimedean field 
 
 
 
 (
 K
 ,
 
 |
 
 ⋅
 
 |
 
 )
 
 
 {\displaystyle (K,|\cdot |)}
 
, the ring of restricted power series tensored with 
 
 
 
 K
 
 
 {\displaystyle K}
 
,

T
          
            n
          
        
        =
        K
        ⟨
        
          ξ
          
            1
          
        
        ,
        …
        
          ξ
          
            n
          
        
        ⟩
        =
        A
        ⟨
        
          ξ
          
            1
          
        
        ,
        …
        ,
        
          ξ
          
            n
          
        
        ⟩
        
          ⊗
          
            A
          
        
        K
      
    
    {\displaystyle T_{n}=K\langle \xi _{1},\dots \xi _{n}\rangle =A\langle \xi _{1},\dots ,\xi _{n}\rangle \otimes _{A}K}

is called a Tate algebra, named for <a href="/facts/John_Tate_(mathematician)/nBntLCG0">John Tate</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> It is equivalently the subring of formal power series 
 
 
 
 k
 [
 [
 
 ξ
 
 1
 
 
 ,
 …
 ,
 
 ξ
 
 n
 
 
 ]
 ]
 
 
 {\displaystyle k[[\xi _{1},\dots ,\xi _{n}]]}
 
 which consists of series convergent on 
 
 
 
 
 
 
 o
 
 
 
 
 k
 ¯
 
 
 
 n
 
 
 
 
 {\displaystyle {\mathfrak {o}}_{\overline {k}}^{n}}
 
, where 
 
 
 
 
 
 
 o
 
 
 
 
 k
 ¯
 
 
 
 :=
 {
 x
 ∈
 
 
 k
 ¯
 
 
 :
 
 |
 
 x
 
 |
 
 ≤
 1
 }
 
 
 {\displaystyle {\mathfrak {o}}_{\overline {k}}:=\{x\in {\overline {k}}:|x|\leq 1\}}
 
 is the valuation ring in the <a href="/facts/Algebraic_closure/yjewG1mM">algebraic closure</a> 
 
 
 
 
 
 k
 ¯
 
 
 
 
 {\displaystyle {\overline {k}}}
 
.
The <a href="/facts/Glossary_of_commutative_algebra/dPjY9Aqe">maximal spectrum</a> of 
 
 
 
 
 T
 
 n
 
 
 
 
 {\displaystyle T_{n}}
 
 is then a <a href="/facts/Rigid-analytic_space/2p3fUBHc">rigid-analytic space</a> that models an affine space in <a href="/facts/Rigid_geometry/2p3fUBHc">rigid geometry</a>.
Define the Gauss norm of 
 
 
 
 f
 =
 ∑
 
 a
 
 α
 
 
 
 ξ
 
 α
 
 
 
 
 {\displaystyle f=\sum a_{\alpha }\xi ^{\alpha }}
 
 in 
 
 
 
 
 T
 
 n
 
 
 
 
 {\displaystyle T_{n}}
 
 by

‖
        f
        ‖
        =
        
          max
          
            α
          
        
        
          |
        
        
          a
          
            α
          
        
        
          |
        
        .
      
    
    {\displaystyle \|f\|=\max _{\alpha }|a_{\alpha }|.}

This makes 
 
 
 
 
 T
 
 n
 
 
 
 
 {\displaystyle T_{n}}
 
 a <a href="/facts/Banach_algebra/3BTHdpg2">Banach algebra</a> over k; i.e., a <a href="/facts/Normed_algebra/RmXrDBHH">normed algebra</a> that is <a href="/facts/Complete_metric_space/lsckmU8G">complete</a> as a <a href="/facts/Metric_space/gnBBRXtc">metric space</a>. With this <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a>, any <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> 
 
 
 
 I
 
 
 {\displaystyle I}
 
 of 
 
 
 
 
 T
 
 n
 
 
 
 
 {\displaystyle T_{n}}
 
 is closed<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> and thus, if I is radical, the quotient 
 
 
 
 
 T
 
 n
 
 
 
 /
 
 I
 
 
 {\displaystyle T_{n}/I}
 
 is also a (reduced) Banach algebra called an <a href="/facts/Affinoid_algebra/2p3fUBHc">affinoid algebra</a>.
Some key results are:

<ul><li>(Weierstrass division) Let 
 
 
 
 g
 ∈
 
 T
 
 n
 
 
 
 
 {\displaystyle g\in T_{n}}
 
 be a 
 
 
 
 
 ξ
 
 n
 
 
 
 
 {\displaystyle \xi _{n}}
 
-distinguished series of order s; i.e., 
 
 
 
 g
 =
 
 ∑
 
 ν
 =
 0
 
 
 ∞
 
 
 
 g
 
 ν
 
 
 
 ξ
 
 n
 
 
 ν
 
 
 
 
 {\displaystyle g=\sum _{\nu =0}^{\infty }g_{\nu }\xi _{n}^{\nu }}
 
 where 
 
 
 
 
 g
 
 ν
 
 
 ∈
 
 T
 
 n
 −
 1
 
 
 
 
 {\displaystyle g_{\nu }\in T_{n-1}}
 
, 
 
 
 
 
 g
 
 s
 
 
 
 
 {\displaystyle g_{s}}
 
 is a unit element and 
 
 
 
 
 |
 
 
 g
 
 s
 
 
 
 |
 
 =
 ‖
 g
 ‖
 >
 
 |
 
 
 g
 
 v
 
 
 
 |
 
 
 
 {\displaystyle |g_{s}|=\|g\|>|g_{v}|}
 
 for 
 
 
 
 ν
 >
 s
 
 
 {\displaystyle \nu >s}
 
.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> Then for each 
 
 
 
 f
 ∈
 
 T
 
 n
 
 
 
 
 {\displaystyle f\in T_{n}}
 
, there exist a unique 
 
 
 
 q
 ∈
 
 T
 
 n
 
 
 
 
 {\displaystyle q\in T_{n}}
 
 and a unique polynomial 
 
 
 
 r
 ∈
 
 T
 
 n
 −
 1
 
 
 [
 
 ξ
 
 n
 
 
 ]
 
 
 {\displaystyle r\in T_{n-1}[\xi _{n}]}
 
 of <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> 
 
 
 
 <
 s
 
 
 {\displaystyle <s}
 
 such that

f
 =
 q
 g
 +
 r
 .
 
 
 {\displaystyle f=qg+r.}
 
<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li>(<a href="/facts/Weierstrass_preparation_theorem/lSP9zffd">Weierstrass preparation</a>) As above, let 
 
 
 
 g
 
 
 {\displaystyle g}
 
 be a 
 
 
 
 
 ξ
 
 n
 
 
 
 
 {\displaystyle \xi _{n}}
 
-distinguished series of order s. Then there exist a unique <a href="/facts/Monic_polynomial/4JzuIJ5R">monic polynomial</a> 
 
 
 
 f
 ∈
 
 T
 
 n
 −
 1
 
 
 [
 
 ξ
 
 n
 
 
 ]
 
 
 {\displaystyle f\in T_{n-1}[\xi _{n}]}
 
 of degree 
 
 
 
 s
 
 
 {\displaystyle s}
 
 and a unit element 
 
 
 
 u
 ∈
 
 T
 
 n
 
 
 
 
 {\displaystyle u\in T_{n}}
 
 such that 
 
 
 
 g
 =
 f
 u
 
 
 {\displaystyle g=fu}
 
.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li>
<li>(Noether normalization) If 
 
 
 
 
 
 a
 
 
 ⊂
 
 T
 
 n
 
 
 
 
 {\displaystyle {\mathfrak {a}}\subset T_{n}}
 
 is an ideal, then there is a finite homomorphism 
 
 
 
 
 T
 
 d
 
 
 ↪
 
 T
 
 n
 
 
 
 /
 
 
 
 a
 
 
 
 
 {\displaystyle T_{d}\hookrightarrow T_{n}/{\mathfrak {a}}}
 
.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li></ul>
As consequence of the division, preparation theorems and Noether normalization, 
 
 
 
 
 T
 
 n
 
 
 
 
 {\displaystyle T_{n}}
 
 is a <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian</a> <a href="/facts/Unique_factorization_domain/aYsX2swC">unique factorization domain</a> of <a href="/facts/Krull_dimension/KQb2uLtz">Krull dimension</a> n.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> An analog of <a href="/facts/Hilbert%27s_Nullstellensatz/TlgoRHuM">Hilbert's Nullstellensatz</a> is valid: the radical of an ideal is the <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a> of all <a href="/facts/Maximal_ideal/nf4DnKVS">maximal ideals</a> containing the ideal (we say the ring is Jacobson).<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<h2 id="results">Results</h2>

Results for polynomial rings such as <a href="/facts/Hensel%27s_lemma/CPCADITS">Hensel's lemma</a>, division algorithms (or the theory of <a href="/facts/Gr%C3%B6bner_bases/efMrjM7w">Gröbner bases</a>) are also true for the ring of restricted power series. Throughout the section, let A denote a linearly topologized ring, separated and complete.

<ul><li>(Hensel) Let 
 
 
 
 
 
 m
 
 
 ⊂
 A
 
 
 {\displaystyle {\mathfrak {m}}\subset A}
 
 be a maximal ideal and 
 
 
 
 φ
 :
 A
 →
 k
 :=
 A
 
 /
 
 
 
 m
 
 
 
 
 {\displaystyle \varphi :A\to k:=A/{\mathfrak {m}}}
 
 the quotient map. Given an 
 
 
 
 F
 
 
 {\displaystyle F}
 
 in 
 
 
 
 A
 ⟨
 ξ
 ⟩
 
 
 {\displaystyle A\langle \xi \rangle }
 
, if 
 
 
 
 φ
 (
 F
 )
 =
 g
 h
 
 
 {\displaystyle \varphi (F)=gh}
 
 for some monic polynomial 
 
 
 
 g
 ∈
 k
 [
 ξ
 ]
 
 
 {\displaystyle g\in k[\xi ]}
 
 and a restricted power series 
 
 
 
 h
 ∈
 k
 ⟨
 ξ
 ⟩
 
 
 {\displaystyle h\in k\langle \xi \rangle }
 
 such that 
 
 
 
 g
 ,
 h
 
 
 {\displaystyle g,h}
 
 generate the <a href="/facts/Unit_ideal/vCvytduX">unit ideal</a> of 
 
 
 
 k
 ⟨
 ξ
 ⟩
 
 
 {\displaystyle k\langle \xi \rangle }
 
, then there exist 
 
 
 
 G
 
 
 {\displaystyle G}
 
 in 
 
 
 
 A
 [
 ξ
 ]
 
 
 {\displaystyle A[\xi ]}
 
 and 
 
 
 
 H
 
 
 {\displaystyle H}
 
 in 
 
 
 
 A
 ⟨
 ξ
 ⟩
 
 
 {\displaystyle A\langle \xi \rangle }
 
 such that

F
 =
 G
 H
 ,
 
 φ
 (
 G
 )
 =
 g
 ,
 φ
 (
 H
 )
 =
 h
 
 
 {\displaystyle F=GH,\,\varphi (G)=g,\varphi (H)=h}
 
.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Bourbaki, N. (2006). Algèbre commutative: Chapitres 1 à 4. Springer Berlin Heidelberg. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783540339373.</li>
<li><a href="/facts/Alexander_Grothendieck/jLQGPY7M">Grothendieck, Alexandre</a>; <a href="/facts/Jean_Dieudonn%C3%A9/gWagzMdu">Dieudonné, Jean</a> (1960). <a href="http://www.numdam.org/item/PMIHES_1960__4__5_0">"Éléments de géométrie algébrique: I. Le langage des schémas"</a>. <a href="/facts/Publications_Math%C3%A9matiques_de_l%27IH%C3%89S/2ur5ofxL">Publications Mathématiques de l'IHÉS</a>. 4. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf02684778">10.1007/bf02684778</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0217083">0217083</a>.</li>
<li>Bosch, Siegfried; Güntzer, Ulrich; Remmert, Reinhold (1984), "Chapter 5", Non-archimedean analysis, Springer</li>
<li>Bosch, Siegfried (2014), <a href="https://books.google.com/books?id=tARYBAAAQBAJ">Lectures on Formal and Rigid Geometry</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783319044170</li>
<li>Fujiwara, Kazuhiro; Kato, Fumiharu (2018), <a href="https://www.maa.org/press/maa-reviews/foundations-of-rigid-geometry-i">Foundations of Rigid Geometry I</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Weierstrass_preparation_theorem/lSP9zffd">Weierstrass preparation theorem</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://ncatlab.org/nlab/show/restricted+formal+power+series">https://ncatlab.org/nlab/show/restricted+formal+power+series</a></li>
<li><a href="http://math.stanford.edu/~conrad/papers/aws.pdf">http://math.stanford.edu/~conrad/papers/aws.pdf</a></li>
<li><a href="https://web.archive.org/web/20060916051553/http://www-math.mit.edu/~kedlaya//18.727/tate-algebras.pdf">https://web.archive.org/web/20060916051553/http://www-math.mit.edu/~kedlaya//18.727/tate-algebras.pdf</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Stacks Project, Tag 0AKZ. <a href="http://stacks.math.columbia.edu/tag/0AKZ" target="_blank">http://stacks.math.columbia.edu/tag/0AKZ</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Grothendieck & Dieudonné 1960, Ch. 0, § 7.5.1. - Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083. <a href="http://www.numdam.org/item/PMIHES_1960__4__5_0" target="_blank">http://www.numdam.org/item/PMIHES_1960__4__5_0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Bourbaki 2006, Ch. III, § 4. Definition 2 and Proposition 3. - Bourbaki, N. (2006). Algèbre commutative: Chapitres 1 à 4. Springer Berlin Heidelberg. ISBN 9783540339373. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Grothendieck & Dieudonné 1960, Ch. 0, § 7.5.3. - Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083. <a href="http://www.numdam.org/item/PMIHES_1960__4__5_0" target="_blank">http://www.numdam.org/item/PMIHES_1960__4__5_0</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Fujiwara & Kato 2018, Ch 0, just after Proposition 9.3. - Fujiwara, Kazuhiro; Kato, Fumiharu (2018), Foundations of Rigid Geometry I <a href="https://www.maa.org/press/maa-reviews/foundations-of-rigid-geometry-i" target="_blank">https://www.maa.org/press/maa-reviews/foundations-of-rigid-geometry-i</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Bosch 2014, § 2.3. Corollary 8 - Bosch, Siegfried (2014), Lectures on Formal and Rigid Geometry, ISBN 9783319044170 <a href="https://books.google.com/books?id=tARYBAAAQBAJ" target="_blank">https://books.google.com/books?id=tARYBAAAQBAJ</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Bosch 2014, § 2.2. Definition 6. - Bosch, Siegfried (2014), Lectures on Formal and Rigid Geometry, ISBN 9783319044170 <a href="https://books.google.com/books?id=tARYBAAAQBAJ" target="_blank">https://books.google.com/books?id=tARYBAAAQBAJ</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Bosch 2014, § 2.2. Theorem 8. - Bosch, Siegfried (2014), Lectures on Formal and Rigid Geometry, ISBN 9783319044170 <a href="https://books.google.com/books?id=tARYBAAAQBAJ" target="_blank">https://books.google.com/books?id=tARYBAAAQBAJ</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Bosch 2014, § 2.2. Corollary 9. - Bosch, Siegfried (2014), Lectures on Formal and Rigid Geometry, ISBN 9783319044170 <a href="https://books.google.com/books?id=tARYBAAAQBAJ" target="_blank">https://books.google.com/books?id=tARYBAAAQBAJ</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Bosch 2014, § 2.2. Corollary 11. - Bosch, Siegfried (2014), Lectures on Formal and Rigid Geometry, ISBN 9783319044170 <a href="https://books.google.com/books?id=tARYBAAAQBAJ" target="_blank">https://books.google.com/books?id=tARYBAAAQBAJ</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Bosch 2014, § 2.2. Proposition 14, Proposition 15, Proposition 17. - Bosch, Siegfried (2014), Lectures on Formal and Rigid Geometry, ISBN 9783319044170 <a href="https://books.google.com/books?id=tARYBAAAQBAJ" target="_blank">https://books.google.com/books?id=tARYBAAAQBAJ</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Bosch 2014, § 2.2. Proposition 16. - Bosch, Siegfried (2014), Lectures on Formal and Rigid Geometry, ISBN 9783319044170 <a href="https://books.google.com/books?id=tARYBAAAQBAJ" target="_blank">https://books.google.com/books?id=tARYBAAAQBAJ</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Bourbaki 2006, Ch. III, § 4. Theorem 1. - Bourbaki, N. (2006). Algèbre commutative: Chapitres 1 à 4. Springer Berlin Heidelberg. ISBN 9783540339373. <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
</ol>

Restricted power series open-in-new

Restricted power series