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<h1>Desvl's blog</h1>
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<p>Mathematics. Articles in English (et en français dans le futur).</p>
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<div class="post-list">
<div class='post-entry content-card'>
<div class="post-entry__header"></div>
<div class="post-entry__content">
<h2 class="post-entry__title">
On the Boyd–Deninger polynomial x+1/x+y+1/y+1, pt. I - The curve
</h2>
In this post we study the Boyd-Deninger polynomial P(x,y)=x+1/x+y+1/y+1. In particular, we are interested in the elliptic curve that is defined by it.
</div>
<div class="post-entry__meta">
<a href="archives/2025/12/" class="post-meta__date button">2025-12-30</a>
<span class="separate-dot"></span><a href="categories/Geometry/" class="button">Geometry</a>
<span class="separate-dot"></span><a href="categories/Geometry/Algebraic-Geometry/" class="button">Algebraic Geometry</a>
<span class="separate-dot"></span><a href="categories/Geometry/Elliptic-Curve/" class="button">Elliptic Curve</a>
</div>
<div class="post-entry__tags"><a href="tags/Elliptic-Curve/" class="post-tags__link button"># Elliptic Curve</a><a href="tags/SageMath/" class="post-tags__link button"># SageMath</a></div>
<a href="2025/12/30/boyd-deninger/" class="post-entry__link">On the Boyd–Deninger polynomial x+1/x+y+1/y+1, pt. I - The curve</a>
</div>
<div class='post-entry content-card'>
<div class="post-entry__header"></div>
<div class="post-entry__content">
<h2 class="post-entry__title">
Boolean ring and algebraic numbers
</h2>
In this post, we study the Boolean ring and see how it can be used in algebraic number theory.
</div>
<div class="post-entry__meta">
<a href="archives/2025/10/" class="post-meta__date button">2025-10-13</a>
<span class="separate-dot"></span><a href="categories/Algebra/" class="button">Algebra</a>
<span class="separate-dot"></span><a href="categories/Algebra/Commutative-Algebra/" class="button">Commutative Algebra</a>
<span class="separate-dot"></span><a href="categories/Number-Theory/" class="button">Number Theory</a>
<span class="separate-dot"></span><a href="categories/Number-Theory/Algebraic-Number-Theory/" class="button">Algebraic Number Theory</a>
</div>
<div class="post-entry__tags"><a href="tags/Exercise-solution/" class="post-tags__link button"># Exercise solution</a><a href="tags/Atiyah-MacDonald/" class="post-tags__link button"># Atiyah-MacDonald</a></div>
<a href="2025/10/13/boolean/" class="post-entry__link">Boolean ring and algebraic numbers</a>
</div>
<div class='post-entry content-card'>
<div class="post-entry__header"></div>
<div class="post-entry__content">
<h2 class="post-entry__title">
Artin-Schreier Extensions
</h2>
We are interested in a special category of field extensions. Let $K$ be a field of characteristic $p \ne 0$, we want to know the structure of an extension of $K$ of degree $p$. It turns out that there lies the an Artin-Schreier polynomial of the form $X^p-X-\alpha$.
</div>
<div class="post-entry__meta">
<a href="archives/2025/05/" class="post-meta__date button">2025-05-16</a>
<span class="separate-dot"></span><a href="categories/Algebra/" class="button">Algebra</a>
<span class="separate-dot"></span><a href="categories/Algebra/Field-Theory/" class="button">Field Theory</a>
<span class="separate-dot"></span><a href="categories/Algebra/Galois-Theory/" class="button">Galois Theory</a>
</div>
<div class="post-entry__tags"><a href="tags/Exercise-solution/" class="post-tags__link button"># Exercise solution</a><a href="tags/Serge-Lang/" class="post-tags__link button"># Serge Lang</a></div>
<a href="2025/05/16/artin-schreier/" class="post-entry__link">Artin-Schreier Extensions</a>
</div>
<div class='post-entry content-card'>
<div class="post-entry__header"></div>
<div class="post-entry__content">
<h2 class="post-entry__title">
Equivalent Conditions of Regular Local Rings of Dimension 1
</h2>
In this post we collect and prove (as detailed as possible) the equivalent conditions of being a Regular local ring of dimension 1.
</div>
<div class="post-entry__meta">
<a href="archives/2025/05/" class="post-meta__date button">2025-05-11</a>
<span class="separate-dot"></span><a href="categories/Algebra/" class="button">Algebra</a>
<span class="separate-dot"></span><a href="categories/Algebra/Commutative-Algebra/" class="button">Commutative Algebra</a>
</div>
<div class="post-entry__tags"><a href="tags/Study-new-content/" class="post-tags__link button"># Study new content</a><a href="tags/Quick-reference/" class="post-tags__link button"># Quick reference</a></div>
<a href="2025/05/11/regular-local-ring/" class="post-entry__link">Equivalent Conditions of Regular Local Rings of Dimension 1</a>
</div>
<div class='post-entry content-card'>
<div class="post-entry__header"></div>
<div class="post-entry__content">
<h2 class="post-entry__title">
The Structure of SL_2(F_3) as a Semidirect Product
</h2>
In this post we determine $SL_2(\mathbb{F}_3)$ using Sylow theory and linear algebra.
</div>
<div class="post-entry__meta">
<a href="archives/2023/11/" class="post-meta__date button">2023-11-11</a>
<span class="separate-dot"></span><a href="categories/Algebra/" class="button">Algebra</a>
<span class="separate-dot"></span><a href="categories/Algebra/Group-Theory/" class="button">Group Theory</a>
</div>
<div class="post-entry__tags"><a href="tags/Exercise-solution/" class="post-tags__link button"># Exercise solution</a><a href="tags/Serge-Lang/" class="post-tags__link button"># Serge Lang</a><a href="tags/Sylow/" class="post-tags__link button"># Sylow</a></div>
<a href="2023/11/11/sl2-f3/" class="post-entry__link">The Structure of SL_2(F_3) as a Semidirect Product</a>
</div>
<div class='post-entry content-card'>
<div class="post-entry__header"></div>
<div class="post-entry__content">
<h2 class="post-entry__title">
A Separable Extension Is Solvable by Radicals Iff It Is Solvable
</h2>
We show that a separable extension is solvable by radical iff it is solvable, i.e. it has a Galois closure with solvable Galois group. The proof is done in a general setting.
</div>
<div class="post-entry__meta">
<a href="archives/2023/10/" class="post-meta__date button">2023-10-21</a>
<span class="separate-dot"></span><a href="categories/Algebra/" class="button">Algebra</a>
<span class="separate-dot"></span><a href="categories/Algebra/Field-Theory/" class="button">Field Theory</a>
<span class="separate-dot"></span><a href="categories/Algebra/Galois-Theory/" class="button">Galois Theory</a>
</div>
<div class="post-entry__tags"><a href="tags/Galois/" class="post-tags__link button"># Galois</a></div>
<a href="2023/10/21/solvable-by-radical/" class="post-entry__link">A Separable Extension Is Solvable by Radicals Iff It Is Solvable</a>
</div>
<div class='post-entry content-card'>
<div class="post-entry__header"></div>
<div class="post-entry__content">
<h2 class="post-entry__title">
Picard's Little Theorem and Twice-Punctured Plane
</h2>
We show that the range of a non-constant entire function's range cannot be a twice-punctured plane.
</div>
<div class="post-entry__meta">
<a href="archives/2023/09/" class="post-meta__date button">2023-09-18</a>
<span class="separate-dot"></span><a href="categories/Analysis/" class="button">Analysis</a>
<span class="separate-dot"></span><a href="categories/Analysis/Complex-Analysis/" class="button">Complex Analysis</a>
</div>
<div class="post-entry__tags"><a href="tags/Walter-Rudin/" class="post-tags__link button"># Walter Rudin</a><a href="tags/Study-new-content/" class="post-tags__link button"># Study new content</a><a href="tags/Analytic-Continuation/" class="post-tags__link button"># Analytic Continuation</a></div>
<a href="2023/09/18/picard-little/" class="post-entry__link">Picard's Little Theorem and Twice-Punctured Plane</a>
</div>
<div class='post-entry content-card'>
<div class="post-entry__header"></div>
<div class="post-entry__content">
<h2 class="post-entry__title">
SL(2,R) As a Topological Space and Topological Group
</h2>
In this post we show that $SL(2,\mathbb{R})$ can be identified as the inside of a solid torus and see what we can learn from it.
</div>
<div class="post-entry__meta">
<a href="archives/2023/08/" class="post-meta__date button">2023-08-12</a>
<span class="separate-dot"></span><a href="categories/Algebra/" class="button">Algebra</a>
<span class="separate-dot"></span><a href="categories/Algebra/Group-Theory/" class="button">Group Theory</a>
<span class="separate-dot"></span><a href="categories/Topology/" class="button">Topology</a>
<span class="separate-dot"></span><a href="categories/Algebra/Lie-Groups/" class="button">Lie Groups</a>
</div>
<div class="post-entry__tags"><a href="tags/Study-new-content/" class="post-tags__link button"># Study new content</a><a href="tags/Lie-theory/" class="post-tags__link button"># Lie theory</a></div>
<a href="2023/08/12/sl2-decomposition/" class="post-entry__link">SL(2,R) As a Topological Space and Topological Group</a>
</div>
<div class='post-entry content-card'>
<div class="post-entry__header"></div>
<div class="post-entry__content">
<h2 class="post-entry__title">
Artin's Theorem of Induced Characters
</h2>
We give a relatively more detailed proof of Artin's theorem in representation theory of finite groups as well as an example of dihedral group.
</div>
<div class="post-entry__meta">
<a href="archives/2023/07/" class="post-meta__date button">2023-07-17</a>
<span class="separate-dot"></span><a href="categories/Algebra/" class="button">Algebra</a>
<span class="separate-dot"></span><a href="categories/Algebra/Representation-Theory/" class="button">Representation Theory</a>
<span class="separate-dot"></span><a href="categories/Linear-Algebra/" class="button">Linear Algebra</a>
<span class="separate-dot"></span><a href="categories/Algebra/Finite-Groups/" class="button">Finite Groups</a>
</div>
<div class="post-entry__tags"><a href="tags/Study-new-content/" class="post-tags__link button"># Study new content</a><a href="tags/Jean-Pierre-Serre/" class="post-tags__link button"># Jean-Pierre Serre</a></div>
<a href="2023/07/17/artin-theorem/" class="post-entry__link">Artin's Theorem of Induced Characters</a>
</div>
<div class='post-entry content-card'>
<div class="post-entry__header"></div>
<div class="post-entry__content">
<h2 class="post-entry__title">
Chinese Remainder Theorem in Several Scenarios of Ring Theory
</h2>
We study the Chinese remainder theorem in various contexts and abstract levels.
</div>
<div class="post-entry__meta">
<a href="archives/2023/05/" class="post-meta__date button">2023-05-27</a>
<span class="separate-dot"></span><a href="categories/Algebra/" class="button">Algebra</a>
<span class="separate-dot"></span><a href="categories/Algebra/Commutative-Algebra/" class="button">Commutative Algebra</a>
<span class="separate-dot"></span><a href="categories/Algebra/Non-commutative-Algebra/" class="button">Non-commutative Algebra</a>
</div>
<div class="post-entry__tags"><a href="tags/Chinese-Remainder-Theorem/" class="post-tags__link button"># Chinese Remainder Theorem</a></div>
<a href="2023/05/27/chinese-remainder-theorem-ring-theory/" class="post-entry__link">Chinese Remainder Theorem in Several Scenarios of Ring Theory</a>
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