Jekyll2022-05-18T22:19:21+00:00https://florianfelix.net/feed.xmlFlorian FelixI am a PhD student in model theory and algebraic geometry at the HHU Düsseldorf in the DFG project GRK2240 under the supervision of Immanuel Halupczok.Florian FelixModel Theory Of Separably Closed Fields2022-05-15T00:00:00+00:002022-05-15T00:00:00+00:00https://florianfelix.net/math/Model-Theory-of-Separably-Closed-Fields<h1 id="model-theory-of-separably-closed-fields">Model Theory of Separably Closed Fields</h1>
<hr />
<p>This blogpost has been a collaboration between myself (Florian Felix) and <a href="https://simoneramello.carrd.co/">Simone Ramello</a>, who wrote the second half about stability theory and separably closed fields with endomorphisms and suggested the idea.
—</p>
<p><em>Separable</em> extensions are the kind of extensions you forget about again after learning about them in the context of Galois theory, if you are not working in an algebra-adjacent field of study or if you prefer fields of characteristic $0$.
However, if you are in positive characteristic the essence is that they are the extensions that are not <em>eldritch horrors that will haunt your dreams</em>, we will elaborate on that and then tell you a bit about how a model-theorist thinks of them:</p>
<h2 id="separable-extensions">Separable Extensions</h2>
<p>Let $K$ be a field and let $L/K$ be a finite field extension. We say that $L/K$ is <em>separable</em> if the minimal polynomial $\mathrm{mipo}_{L/K}(x)$ is separable, which is obbviously a stupid-sounding definition, but a polynomial $f$ is separable, if the set of zeros over the algebraic closure of the ambient field has cardinality $\deg(f)$ or in other more fany words if it has no double roots in any field extension.</p>
<p>In characteristic $0$ this is a boring definition:</p>
<hr />
<p><strong>Definition.</strong> Let $K$ be a field and $\mathrm{char}(K)=0$ and let $f \in K[X]$ be irreducible, then $f$ is separable.</p>
<hr />
<p><strong>Proof:</strong>
We can assume that $\deg(f)\not=0,1$ in which cases its obvious.
We assume that $f$ has a double root $\alpha$, then we can write $f=(X-\alpha)^2*g$ and form the derivative with the product rule:
\(f'=2*(X-\alpha)*f+(X-\alpha)^2*f'\)
which shows that $\alpha$ is also a root of $f’$. But because we are in characteristic $0$ its always true that $\deg(f’)=\deg(p)-1$, so $p’$ is not a constant polynomial.
But since $f$ and $f’$ have the same root $\mathrm{gcd}(f,f’) \not=1$ and because of irreducibility of $f$ this implies $f \mid f’$ contradicting their degree.</p>
<p>The proof also suggests which polynomials make problems in the case of positive characteristic. If $\mathrm{char}(f)=p$, then for instance $X^p-X-a$ might be irreducible, but has constant derivative.</p>
<h2 id="separably-closed-fields">Separably Closed Fields</h2>
<p>Algebraically closed fields are maximal algebraic extensions, the same philosophy applies for separably closed fields.</p>
<hr />
<p><strong>Definition.</strong> A field $K$ is called <em>separably closed</em> if there is no proper separable extension $L/K$.</p>
<hr />
<p>The proof of the existence of separable closures of fields is not unlike the one for algebraically closed fields and in a similar manner the separable closure of a field is essentially unique.
Again, in characteristic $0$ this notion coincides with being algebraically closed.</p>
<h2 id="ah-perfection">Ah, perfection</h2>
<p>It is not clear to me what makes them so perfect, but rather they are not <em>hideous</em>:</p>
<hr />
<p><strong>Definition.</strong> A field $K$ is called <em>perfect</em> if $K^\mathrm{sep}=K^\mathrm{alg}$. If $\mathrm{char}(K)=p$ and $K$ separably closed, we measure how far it is away from perfection via the <em>imperfection degree</em> or <em>Ershov invariant</em> $e$, defined via $p^e=\deg(K/K^p)$ - if $\mathrm{deg}(K/K^p)$ is finite and is $\infty$ otherwise.</p>
<hr />
<p>In more generality we call a finite set ${a_1, \ldots,a_n} \subseteq K$ $p$-independent, if the corresponding <em>$p$-monomials</em> are $K^p$-linearly independent, where a $p$-monomial is of the form
\(\prod a_{i}^{e_i}\)
for $0 \leq e_i < p$. An arbitrary set is $p$-independent if every finite subset is $p$-independent. A $p$-basis is a maximally $p$-independent set and to no surprise the imperfection degree coincides with the cardinality of a $p$-basis. We will only be interested in the case where $e$ is finite.</p>
<h2 id="model-theory">Model Theory</h2>
<p>Now we want to study separably closed fields in accordance to their first-order theory. It is not hard to see that we can axiomatize the theory of separably closed fields of characteristic $p$ and imperfection degree $e \in \mathbb{N}$ in the language of rings \(\mathcal{L}_{\text{ring}}:=\{0,1,+,\cdot\}\)
by describing that every separable non-constant polynomial has a root and that there are $p$-independent sets of cardinality up to $e$. Sadly the pure language of rings is suboptimal for understanding this theory, thus we need to enrich the language a little bit:
\(\mathcal{L}_{p,e}:=\mathcal{L}_{\text{ring}} \cup \{a_1, \ldots, a_e\} \cup \{\lambda_{\sigma} \mid \sigma \in (p^{e})^{< \omega}\}\)
where $a_i$ are constant symbols and denote a $p$-basis and the $\lambda_{\sigma}$ are unary function symbol and $(p^e)^{<\omega}$ stands for finite tuples of elements ${0, \ldots, p^e-1}$ and to interpret them we consider the $p$-monomials $m_{0}, \ldots,m_{p^e-1}$ generated by the $p$-basis ${a_1, \ldots,a_e}$, if $x$ is now any element of the ambient field we can write
\(x=x_{0}^p m_0 + \ldots, x_{p^e -1}^p m_{p^e -1}\)
now for instance $\lambda_{j}=x_j$ where $j$ is interpreted as the tuple of length 1, that contains $j \in{0, \ldots,p^e -1 }$, now we can iterate the process and do the same for $x_{j}$ instead of $x$ an get $\lambda_{j,i}(x)=\lambda_{i}(\lambda_{j}(x))=(x_j)_i$ and iterate this to get the interpretation of the $\lambda$-functions for arbitrary finite tuples.</p>
<p>We denote the theory of separably closed fields of characteristic $p$ and imperfection degree $e \in \mathbb{N}$ in
this language by $\mathrm{SCF}_{p,e}$. It admits quantifier elimination:</p>
<hr />
<p><strong>Theorem.</strong> (Yuri Ershov) $\mathrm{SCF}_{p,e}$ is model-complete.</p>
<hr />
<p>The proof only uses a few facts about fields of definition and general varieties and enables us to do the following.</p>
<hr />
<p><strong>Theorem.</strong> (Delon, in a more general setting) $\mathrm{SCF}_{p,e}$ admits quantifier elimination.</p>
<hr />
<p><strong>Proof:</strong> It suffices to prove that for a $\aleph_1$-saturated model $L$ and a countable model $K$ and $A \leq K$ and $B \leq L$, if $A \cong B$, this isomorphism extends to an elementary embedding of $K$ into $L$.
Here is how the added $\lambda$-functions save the day: Because every substructure needs to contain all coordinates $x_{\sigma}$ for a given $x$ we see that for a given substructure $k$ of a model of $\mathrm{SCF}_{p,e}$ the substructure has imperfection degree $e$: It has to be at least $e$ since ${a_1, \ldots,a_e} \subseteq k$ and it is at most $e$ because all the coordinates appear due to the $\lambda$-functions witnessing independence.
Now this is easy, let $k_1$ be a substructure of $K$ and $k_2$ a substructure of $L$ with $k_1 \cong k_2$, then we can easily extend this isomorphism to the separable closure of their fraction fields, which have to have the same imperfection degree like the ambient structure and then we complete with model completeness and saturation of $L$.</p>
<h2 id="the-stable-fields-conjecture">The stable fields conjecture</h2>
<p>If you are a model theorist, and you are given a theory, the first question you might come up with is: How many models are there? Or more precisely, if you give me a cardinal, how many models (up to isomorphism) are there of that cardinality? The question is somehow natural – classifying structures down to some decent notion of <em>equality</em> is common throughout mathematical fields (pun not intended), and in a sense one would like to know exactly how extensive the library of possible structures to pick from is. As natural as it is, it ends up being a highly complicated investigation; for a cardinal $\kappa$, there are at most $2^\kappa$ models of cardinality $\kappa$, but the whole spectrum between $1$ (so-called $\kappa$-categoricity) and $2^\kappa$ is a priori possible. The next natural step, then, is to reduce the class of theories we look at: we want <em>tame</em> theories.</p>
<hr />
<p>Definition<sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup>. A theory $T$ is ($\kappa$-)stable if, for any model $M \vDash T$ and $A \subseteq M$ of cardinality at most $\kappa$, the set $S_1(A)$ of $1$-types over $A$ has size at most $\kappa$.</p>
<hr />
<p>This admits a very nice rephrasing in terms of the <em>omission</em> of certain combinatorial patterns, under the belief that one might obtain tameness (or, in other words, remove wildness) for theories by prescribing that their definable sets avoid certain infamous (as the kids say, <em>cursed</em>) combinatorial configurations. This philosophy guides the efforts in <em>abstract</em> (or <em>pure</em>) model theory, leading to the identification of classes of theories like
\(\mathrm{NIP}\) theories, \(\mathrm{NIP}_{n}\) theories, \(\mathrm{NTP2}\) theories and so on. These combinatorial definitions allow us to split the landscape of mathematical theories into chunks, a bit like a map, in which we can place the theories we know according to their <em>complexity</em>. Before moving on to see how this relates to our favourite fields, let us see an example.
First of all, what is the combinatorial pattern omitted in stability? One wants to avoid the presence of a linear order (definable) in the structure, to avoid the insurgence of too many cuts (and hence too many types, and hence too many models). We will say that a theory $T$ has the order property (OP) if there is a formula $\phi(x,y)$ and a model $M \vDash T$ and elements \((a_i)_{i \in \mathbb{N}}\), \({(b_j)}_{j \in \mathbb{N}}\) in $M$ such that $M \vDash \phi(a_i,b_j)$ if and only if $i \leq j$. A theory with the order property encodes the order $(\mathbb N, \leq)$, which is a red flag – as said before, this determines cuts, and that is too high a complexity. We will then say that $T$ is <em>stable</em> if it doesn’t have the order property.</p>
<p><strong>Example.</strong> The structure $(\mathbb Z, +, \cdot, 0, 1)$ (affectionately known as <em>the Prince of Darkness</em>, or <em>Satan</em>), or rather its theory, has the order property and is highly unstable. To see why, remember that by the so-called Lagrange’s Four Squares Theorem, every natural number can be written as the sum of four squares of integers. In other words, we can define $\mathbb N \subset \mathbb Z$ by saying that $x \in \mathbb N$ if and only if $\exists x_1, x_2, x_3, x_4 (x = x_1^2+x_2^2+x_3^2+x_4^2)$. As we have defined the set of positive elements, we have effectively defined the linear order $\leq$ on $\mathbb Z$, and hence proved that this theory is not stable.</p>
<p>What about fields? Well, there are good news.</p>
<hr />
<p><strong>Theorem.</strong> (Carol Wood) The theory $\mathrm{SCF}_{p,e}$ is stable.</p>
<hr />
<p>The proof is achieved with the help of quantifier elimination through a <em>third</em> characterization of stability – namely <em>counting types</em>. As with models, one says that $T$ is ($\kappa$-)stable if, for any model $M \vDash T$ and $A \subseteq M$ of cardinality $\kappa$, the set $S_1(A)$ of $1$-types over $A$ has size at most $\kappa$. We have thus seen that $\mathrm{SCF}_{p,e}$ is a relatively tame theory, and in fact it is in a sense the only known theory of fields which happens to be stable (algebraically closed fields are also stable, in fact being algebraically closed is equivalent to being $\omega$-stable for a field!).</p>
<p><strong>Conjecture.</strong> Any stable infinite field is separably closed.</p>
<p>We are quite far from seeing a definite answer for this conjecture, yet all evidence (e.g. work on the henselianity conjecture, or on its cousin, Shelah’s conjecture around NIP fields) seem to indicate a positive answer to the so-called <em>stable fields conjecture</em>.</p>
<h2 id="enriching-the-structure">Enriching the structure</h2>
<p>Separably closed fields are all nice and well-understood, to the point that their theory sits comfortably in the paradise of stable theories (where almost all machinery seems to work as it should). One might then be tempted to add a little spice to their structure; we have already seen an enrichment of the original pure ring structure, via $\lambda$-functions – this is however, in a sense, an harmless expansion. It is an expansion by definable objects, and while naming them gives us QE, it is not a very natural expansion, where ‘natural’ might be defined as ‘met in their daily life by the average mathematician’. One might want to enrich this structure further, by adding some more interesting objects, like endomorphisms.</p>
<p>This is per se not a complicated recipe – it is just the addition of a symbol $\sigma$ for some map of rings $\sigma: K \to K$ to our language. The theory will then have to also mention $\sigma$, and the fact that it is a map. All of this is straightforward.
What happens next, however, is highly complicated. The resulting theory, or rather its model companion \(\mathrm{SCFE}_{e}\), is terribly non-trivial, and while it is somehow building up on the stability of $\mathrm{SCF}_{p,e}$ (most proofs rest on the shoulders on the machinery of canonical bases), it is fundamentally more complex.</p>
<hr />
<p><strong>Theorem.</strong> (Zoé Chatzidakis, Ehud Hrushovski) Consider the language $\mathcal{L}_1$ obtained from the language of rings expanded with a symbol $\sigma$ for an endomorphism and with countably many $n$-ary relation symbols $R_n$ for $\sigma(K)$-linear independence in $K$. The model companion of the $\mathcal{L}_1$-theory of separably closed fields of degree of imperfection at most $e$ with an endomorphism $\sigma$ is axiomatized by:</p>
<ol>
<li>$K$ is a separably closed field, $\sigma: K \to K$ is an endomorphism,</li>
<li>there is some $x$ such that $R(1,x)$,</li>
<li>a confusing axiom, called the <em>genericity</em> axiom: if $U,V$ are varieties defined over $K$, of the same dimension, such that $V \subseteq U \times U^\sigma$ projects dominantly onto $U$ and $U^\sigma$, and moreover if $(a,b)$ is a generic of $V$ then $a \in K(b^{1/q})^{\mathrm{sep}}$ for some $q = p^n$, where $p = \mathrm{char}(K)$, then there is $a \in K$ such that $(a,\sigma(a)) \in V$.</li>
</ol>
<hr />
<p>The last axiom is very weird, and at first doesn’t seem intuitive at all, yet it is exactly what is needed to ensure the sufficient <em>richness</em> that makes this theory the model companion of the natural theory of separably closed fields with an endomorphism. In fact, one might argue that this theory is very natural in the following sense: it describes the asymptotic behaviour of the Frobenius on a separably closed fields. More precisely, take the set $Q$ of prime powers. If $q = p^n \in Q$, let $F_q$ be a separably closed (but not algebraically closed!) field of characteristic $p$ together with $\sigma_q(x) = x^q$. Let $F^* = \prod_{q \in Q} (F_q,\sigma_q)/\mu$ be the ultraproduct of these fields with respect to a non-principal ultrafilter $\mu$ on $Q$. Then,</p>
<p><strong>Theorem.</strong> (C., H.) $F^*$ is a model of $\mathrm{SCFE}_e$ for some $e$.</p>
<h2 id="literature-and-further-reading">Literature and further reading</h2>
<ul>
<li>Chatzidakis, Z. & Hrushovski, E. (2004). Model theory of endomorphisms of separably closed fields. Journal of Algebra - J ALGEBRA. 281. 567-603.</li>
<li>Messmer, M. (2017). Some Model Theory of Separably Closed Fields. In D. Marker, M. Messmer, & A. Pillay (Authors), Model Theory of Fields (Lecture Notes in Logic, pp. 135-152). Cambridge: Cambridge University Press. doi:10.1017/9781316716991.005</li>
<li>Ershov, Y., Fields with a solvable theory. Doklady Akadeemii Nauk SSSR, vol.174 (1967), pp.19-20; English translation, Soviet Mathematics, vol.8 (1967), pp.575-576.</li>
<li>Tent, K., & Ziegler, M. (2012). A Course in Model Theory (Lecture Notes in Logic). Cambridge: Cambridge University Press. doi:10.1017/CBO9781139015417</li>
</ul>
<div class="footnotes" role="doc-endnotes">
<ol>
<li id="fn:1" role="doc-endnote">
<p>This section previously contained an incorrect definition, we thank Artem Chernikov for pointing it out. <a href="#fnref:1" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
</ol>
</div>Florian FelixModel Theory of Separably Closed FieldsAdphd2022-05-11T00:00:00+00:002022-05-11T00:00:00+00:00https://florianfelix.net/personal/ADPhD<h1 id="adphd">ADPhD</h1>
<p>The first months of my PhD were not easy. I had trouble concentrating on all the long papers I had to read, I could not get anything really <em>done</em> and due to some additional emotional stress in my personal life these pesky things names <em>executive functions</em> really did not want to function at all. I would stare at the screen at times, not getting anything done. This has happened before. I would always stay up late until my brain decided to really focus on it for hours, but having a new environment and a lot less schedule and communication made everything so much worse.
I have a few friends who were diagnosed with ADHD, not so much earlier and it all sounded very much relatable. I always thought “Well, I am certainly a bit on the spectrum, but it would not change anything anyway.” until my problems got too big and I felt way too bad way too often. I made an appointment with a specialist.</p>
<h2 id="diagnosis">Diagnosis</h2>
<p>I filled out a long paper with questions, had an hour of chat with the specialist and he told me “Well, you have ADHD, we can try medicaments and I recommend it.” I asked questions how dangerous this would be, how addicting they are and was reassured that it was all relatively fine. So I agreed.</p>
<h2 id="first-days">First Days</h2>
<p>It is hard to describe what the meds actually do, because executive function is so hard to explain. I realized very quickly that it could not have been a misdiagnosis (supposedly neurotypical people just get really nervous and cocky from taking them), my memory was working faster. I could pay attention to talks, I was only <em>mildly</em> interested in and I had the same ideas, but I could write down tedious details which revealed problems much faster. It felt and feels like someone started to organize this mess of a brain a little bit.
The weirdest thing for me was to realize all sorts of stuff I previously had a lot of problems with vanished or got much more bearable and then realized after a bit of googling that that was also a typical symptom of ADHD, for instance emotional responses to rejection (or everything that maybe could look like rejection if you look at it through a distorted mirror and are a little bit drunk) were much more coherent. It was and is easier to follow the rational explanation, instead of going in circles about how bad everything is.</p>
<h2 id="conclusion-1">Conclusion 1</h2>
<p>If you wear glasses, you probably know the feeling of not realizing how bad something was until it gets a bit better. I recently got glasses and still can not realize how entirely shitty the world looks if I take them off. Nobody complains about shortsighted people not seeing very well when they do not have their glasses. And it seems the chemistry of my brain is slightly off and while I have to live with this fact and not everything suddenly got solved - for instance I can not take theses meds everyday, I should for instance take breaks at the weekend and they do not carry over the whole “being awake” period of the day - it is very good to know that this is the case.</p>
<h2 id="conclusion-2">Conclusion 2</h2>
<p>Like mentioned in the first conclusion, it was not clear to me how bad it was. I now feel like I have time to do at least a fraction of all the things I want to do, because I do not waste that much time staring at walls conjuring up the very concept of <em>doing something</em> instead of nothing. And a lot I was very ashamed of, was at least partially not my fault. If this sounds relatable, then you try and should get checked. There is a lot of undiagnosed ADHD in relatively high-functioning adults. I should also mention that it almost certainly helped, that I am perceived as male by doctors as female friends had a lot harder time finding someone willing to diagnose them or prescribe medication. This should however just acknowledge this fact and not discourage you. Being neuroatypical is not a shame. Also note that I am not an expert, this is purely meant as to document my experience and maybe in the way make more people aware.</p>Florian FelixADPhD The first months of my PhD were not easy. I had trouble concentrating on all the long papers I had to read, I could not get anything really done and due to some additional emotional stress in my personal life these pesky things names executive functions really did not want to function at all. I would stare at the screen at times, not getting anything done. This has happened before. I would always stay up late until my brain decided to really focus on it for hours, but having a new environment and a lot less schedule and communication made everything so much worse. I have a few friends who were diagnosed with ADHD, not so much earlier and it all sounded very much relatable. I always thought “Well, I am certainly a bit on the spectrum, but it would not change anything anyway.” until my problems got too big and I felt way too bad way too often. I made an appointment with a specialist.An Outrageous Proof2022-02-17T00:00:00+00:002022-02-17T00:00:00+00:00https://florianfelix.net/math/An-outrageous-proof<h2 id="an-absurd-proof-that-sqrt2-is-irrational">An absurd proof that $\sqrt{2}$ is irrational</h2>
<p><em>Edit:</em> I was made aware that this proof appeared on <a href="https://math.stackexchange.com/a/1311239/510814">Stackexchange</a> before and I don’t want to claim authorship at all!</p>
<p>Sometimes a good tool to make people remember something is to do something utterly outrageous. I will hence give you the worst proof you have ever seen to show that $\sqrt{2}$ is irrational. It uses a bit of non-trivial number theory and a bit of model theory and also it uses a weak form of the axiom of choice.</p>
<h3 id="ultraproducts">Ultraproducts</h3>
<p>Ultraproducts pose a fundamental tool in model theory and we will do some unholy work with them. First a quick reminder: We fix some language $\mathcal{L}$ and take any index set $I$ to index a bunch of $\mathcal{L}$-structures $(M_i)_{i \in I}$. We then take a non-trivial ultrafilter <sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup> $\mathcal{U}$ of $I$ and define</p>
\[\prod_{\mathcal{U}} M_i = \prod_{i \in I} M_i / \sim_{\mathcal{U}}\]
<p>where \((x_i)_{i \in I} \sim (x'_i)_{i \in I}\) if and only if the set
\(\{i \in I \mid x_i=x'_i \}\) is in $\mathcal{U}$.</p>
<p>One then also interprets $\prod_{\mathcal{U}} M_i$ as an $\mathcal{L}$-structure by checking that functions remain well-defined and by defining relations to hold iff they hold on filter sets. This leads to the following result:</p>
<blockquote>
<h4 id="łośs-theorem">Łoś’s theorem</h4>
<p>Let $\varphi$ be any $\mathcal{L}$-sentence - for instance if \(\mathcal{L}=\{0,1,+,-,\ast\}\) is the language of rings $\varphi$ could be the sentence that says “$2$ is a square” - then
\(\prod_{\mathcal{U}} M_i \vDash \varphi \Leftrightarrow \{i \in I \mid M_i \vDash \varphi \} \in \mathcal{U}.\)</p>
</blockquote>
<p>So the ultraproduct believes a sentence to be true if and only if the set of models that believe the sentence is a filter set.</p>
<h3 id="squares-in-finite-field">Squares in finite field</h3>
<p>If $p=8n+3$ is prime for some $n \in \mathbb{Z}$, then we can use basic properties of the <a href="https://en.wikipedia.org/wiki/Legendre_symbol">Legendre Symbol</a> to see, that</p>
<p>\(\left(\frac{2}{p}\right)=-1\)
which is equivalent to say that $2$ is not a square modulo $p$.</p>
<p>By a bit more heavy machinery, namely <a href="https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions">Dirichlet’s theorem on arithmetic progression</a>
there are infinitely many primes of the form $p=8n+3$.
Let $P$ be the set of those primes.</p>
<h3 id="the-ridiculous-magic-trick">The ridiculous magic trick</h3>
<p>We now envoke the axiom of choice<sup id="fnref:2" role="doc-noteref"><a href="#fn:2" class="footnote" rel="footnote">2</a></sup> to produce a non-principal ultrafilter on $P$ and build a new field:
\(F:=\prod_{\mathcal{U}} \mathbb{F}_p.\)
By Łoś’s theorem the following things are true:</p>
<ul>
<li>$F$ is a field, cause all the $\mathbb{F}_p$ are fields and that is a first-order property.</li>
<li>All the $\mathbb{F}_p$ believe that there is no square root of $2$, so $F$ also does not have a square root of $2$.</li>
<li>There are arbitrarily large $p$’s in $P$, hence the theorem $\underline{1+\ldots+1}_{p-\text{times}} \neq 0$ is true in $\mathbb{F}_p$ for cofinitely many $p$, hence $F$ necessarily has characteristic $0$.</li>
<li>Because $F$ has characteristic $0$, there is a unique embedding $\mathbb{Q} \hookrightarrow F.$</li>
<li>So $\mathbb{Q}$ can not have a square root of 2, because $F$ does not even have one!</li>
</ul>
<p>I wonder if there is an easier proof for this…</p>
<div class="footnotes" role="doc-endnotes">
<ol>
<li id="fn:1" role="doc-endnote">
<p>A non-trivial ultrafilter is a maximal filter that is not a principal ultrafilter (i.e. just all the sets that contain some fixed $x \in I$) <a href="#fnref:1" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:2" role="doc-endnote">
<p>Indeed something a tiny bit weaker! <a href="#fnref:2" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
</ol>
</div>Florian FelixAn absurd proof that $\sqrt{2}$ is irrationalA Week Of Mathematical Outreach, The Good, The Ba(r)d And The Ugly2022-02-09T00:00:00+00:002022-02-09T00:00:00+00:00https://florianfelix.net/math/A-week-of-mathematical-outreach,-the-good,-the-ba(r)d-and-the-ugly<h2 id="a-week-of-mathematical-outreach-the-good-the-bard-and-the-ugly">A week of mathematical outreach, the good, the ba(r)d and the ugly</h2>
<p>I like to talk about my research. I really do, even with people, who are not model theorists or not even mathematicians. It is fun to make up more or less <em>unhinged</em> ways of describing what you do and trying to find metaphors for very abstract thinking. So I reached out and asked whether I could be a curator for the German version of <a href="https://twitter.com/realsci_DE">RealScientists</a>. I followed them for a while and liked their approach of just letting scientists who are not all specialized into scicomm talk about what they do.
About a week before my turn was due, I realized:</p>
<p><strong>I would be the first mathematician on that account.</strong></p>
<p>And I am not primarily a scicomm person. I did a lot of teaching before I got a fancy PhD position that has no teaching obligations, but I have no actual formal training. So I decided to do what I can do best: <em>Improvise</em> and <em>Imitate</em>.
It is hard to actually ascertain how well that worked out, but let me tell you first what I actually did.</p>
<h3 id="the-facts">The Facts</h3>
<p>Personally, I am sometimes irritated by the way mathematical outreach is <em>often</em> practiced. While e.g. a biologist tells you what they do, but will not go into the technical details and all the pesky stuff, my impression was that mathematicians <em>tend</em> to rather talk about basics that might have nothing to do with their research at all. And I get, why that is the case. Mathematics uses its own language with little to no actual representation in the real world. Our whol shtick is abstraction and while every abstraction had a practical idea somewhere in the history of its creation, this might be \(\aleph_0\) iterations ago and not useful anymore at all. I also did that, but I tried to also talk about things related to my research. And while there are things that are worse, I assume that model-theoretic motivic integration is very high on the scale of iterations of abstractions needed to understand why the hell we would be interested.</p>
<p>I wanted to start with things that might be known to people, who are into popsciency math. The first thing I did was a brief <em>Rundumschlag</em> (just to confer with the comment of a friend that Germans will just use words in their mother tongue, if there is nothing as cool in English), tell people that I do “ping-pong” between semantics and formalism. Drop a few terms that might become relevant. I told them they could ask any questions and asked them what they were interested in in particular. I gave a threaded proof of the n=4 case of Fermats Last Theorem - because someone mentioned Fermat and I figured that would not be too bad.</p>
<p>In my freetime, I feel mostly like a storyteller and tried to this week, so I created a couple of threads that had a narrative</p>
<ul>
<li>The betrayal of the integers, in which I told about the structures we learn about in roughly the order of their inclusions \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}\) and where most students would find them increasingly scary. Everyone gets the natural numbers, but <strong>what the fuck is that i business</strong> and that stuff. And after I got to $\mathbb{C}$ I rolled it back up and told them that from the point of view of a model theorist, its the other way around. The complex numbers are tame and the further you go down the more unhinged nonsense you get. Of course that leads to rich mathematical structure, but combinatorical it gets worse and worse - apart from maybe the step from $\mathbb{Z}$ to $\mathbb{N}$ which are arguably equally bad. Of course everything in the context of a language, where we can talk about addition and multiplication.</li>
<li>Why Gödels Incompleteness theorems are less astonishing than what is sometimes communicated, but still very great.</li>
<li>Why I like the completeness theorem and compactness a lot more. <em>duh</em></li>
<li>What is a <em>valued</em> field and what are their geometrical properties.</li>
<li>What did I do in my bachelor thesis and how did I find that topic. A bit about how $(a+b)²=a²+b²$ is not as stupid as you were told in school. That there are other characteristics than $0$ and why inseparable polynomials worry us.</li>
<li>What did I do in my master thesis and how did I get there. What did I do in Oxford and why are local-global principles interesting?</li>
<li>A very rough sketch on what interests me in my own research now.</li>
</ul>
<p>Apart from that I did do a few interactive posts like “Do you also like to use chalkboards in your work?” or “What kind of software do you use to write?” and ended up with a few things about my personal life like that I try to not work too much on weekends and play a lot of tabletop rpgs. I also did a post, which was very important to me, how math is hard and not hard at the same time, elaborating that you do not need to be a (white cis-male) genius to be a mathematician, but that its also valid to struggle and most of the people calling everything trivial struggled themselves. I also gave them a little semimathematical riddle.</p>
<h3 id="the-good">The Good</h3>
<p>I had the impression that people were generally interested. There were a lot of questions. People really liked my story about how the integers came to betray us in the end (or maybe more the beginning) of our mathematical journey. Generally the more narrative elements were well received, though of course I do not know whether it led to deeper understanding or not (that might very well not be the goal at all). People also liked the more sociological comments about math, they also liked to try and riddle with me.</p>
<h3 id="the-bard">The Ba(r)d</h3>
<p>I feel with the more complex math I lost the narrative structure a bit. It would have been better if I made up as elaborate stories for why we would be interested in valued fields as to why the integers betrayed us. This is harder, I think. To me it still seems like a good idea. It was a mistakte to try and explain the proof for Fermats Last Theorem for $n=4$. The proof does of course not need any sophisticated math, but I underestimated the length of the argument before I started typing it. I don’t think there was much gained from explaining this. I thought it would lead to a good narrative, because I wanted to explain how weird the integers were, but in the end this could have also been done by just mentioning how incredibly difficult the proof for the general case was. I also don’t think I was able to convey in what niche I am actually invested. I talked a bit about motivic integration, but I still struggle to explain to mathematicians what this is good for - it was not easier to do this for people without that training.</p>
<h3 id="the-ugly">The Ugly</h3>
<p>My usual twitter account has about 1800 followers by now. That is also not a small number, but they are filtered by an extensive blocklist, by the occasional weird surreal joke and by a giant banner that pretty much should scare of any random libertarian. That account had 16000 and no such filters. The highlights of weird comments included:</p>
<ul>
<li>Someone calling me backwards for not wanting to abolish all theoretical computer science in German schools (not that I have expertise on that and I was careful with my wording, but… what?)</li>
<li>Someone calling me backwards for using chalkboards (and promoting smartboards with handwriting recognition, because humans might not be able to see the difference between an i, a j and so on … but the algorithm definitely can do that)</li>
<li>A few typical complaints that all funding for math should just go into cancer research</li>
</ul>
<h3 id="conclusion">Conclusion</h3>
<p>Well, I don’t know! I think I did not too badly, but I do not have any data on how well I was actually able to communicate what I do. I think narrative structures are a good thing to help people navigate through our abstract nonsense. I also gained an enormous amount of respect for people who actually do mathematical scicomm.</p>Florian FelixA week of mathematical outreach, the good, the ba(r)d and the ugly I like to talk about my research. I really do, even with people, who are not model theorists or not even mathematicians. It is fun to make up more or less unhinged ways of describing what you do and trying to find metaphors for very abstract thinking. So I reached out and asked whether I could be a curator for the German version of RealScientists. I followed them for a while and liked their approach of just letting scientists who are not all specialized into scicomm talk about what they do. About a week before my turn was due, I realized: