In fact, if $E/\Bbb F_p$ is an infinitary extension then $E$ is union of infinite copies of $\Bbb F_i$s. In that case, $E^\times$ is not finitely generated.
@BalarkaSen Here's a streamlined argument. Let $F$ be an algebraic extension of $F_p$ with $F^\times$ finitely generated. Then if $F$ is infinite, $F^\times$ has a free part, i.e., an element of infinite order. But that's silly; because call such an element $a$; then $a$ is algebraic over $F$, this $F(a) / F$ is a finite extension, thus $F(a)$ is finite, thus $a$ has finite order.
A few months ago I was thinking all fields. Even trying to prove Sylow by Galois theory. A few weeks ago I was thinking all groups, trying to construct Galois theory in groups. Now while reading Atiyah-McDonald I am all rings. Finitely generated rings everywhere.
@BalarkaSen Prove or disprove the following: if a graph $G$ has a hamiltonian path between any pair of distinct points $(a,b)$, then it has a hamiltonian cycle.
@BalarkaSen I've never heard of it, but if you take pretty much any combination of math words and string them together i'm sure some mathematician out there has written 100 papers about it
@GustavoMontano It's just that give a polynomial you can form a group of symmetries out of them. Take $x^2 - 2$ over $\Bbb Q$, for examples. You can form another set $F$ from that with elements of the form $a + b\sqrt{2}$ for $a, b \in \Bbb Q$. Note that you can do addition multiplication subtractions and inversions in $F$.
The "galois group" is the maps $F \to F$ preserving addition and multiplication which sends rationals to rationals. in that way, $f(a + b\sqrt{2})$ goes to $f(a) + f(b) f(\sqrt{2}) = a + b f(\sqrt{2})$ as $a, b \in \Bbb Q$. But then $f(\sqrt{2}^2 - 1) = f(0) = 0$, and $f(\sqrt{2}^2 - 1) = f(\sqrt{2}^2) - f(1) = f(\sqrt{2})^2 - 1 = 0$, so $f(\sqrt{2})$ cam be either $\sqrt{2}$ or $-\sqrt{2}$
So you have two maps, one sending $a + b\sqrt{2}$ to $a + b\sqrt{2}$ another sending $a + b\sqrt{2}$ to $a - b\sqrt{2}$. Call these maps $f$ and $g$ resp. Note that applying $g$ twice to $a + b\sqrt{2}$ gives $a + b\sqrt{2}$ back, so $g^2 = f$.
In this sense, $\{f, g\}$ forms a group, fancy name being "cyclic group of order $2$" and fancy notation being $\Bbb Z_2$.
this is the corresponding galois group for $x^2 - 2$.
so in other words it's a way of quantifying the statement that $\sqrt{2}$ and $-\sqrt{2}$ are algebraically interchangable, from the point of view of rational numbers.
Oh ... Tried a few. Nothing's ever been as good as the Parker. Even a new, more expensive Parker. Or Montblanc :( @Pedro is also a fountain pen fanatic!
What I'm basically wondering is if there's a way to show that something can be "evaluated globally" like the way you can take the integral of a huge portion of a function without moving across the entire function piece by piece (my notions of time complexity come from programming so hopefully that makes sense)
I basically have a sum and am wondering if it can be shown that it is or isn't possible to evaluate it without walking over every step if there's information I'm willing to discard.
And I'm also curious if there is a notion of this distinction I have in my head that some operations can be done "with just a function and a range" rather than by individually evaluating the function at points within the range.
The original context I was wondering about was lattice points
it seems like a lattice point intersection in a function is a very "local" property of the function
I haven't been able to find a way to evaluate something like "does this function have any lattice points" without actually checking each lattice point in a range
@0x5f3759df This sounds very broad. I'm not sure how to approach something like this. It would be useful if you could narrow the scope of the problem. By the way, what does it mean for a function to have "lattice points"?
I'm using "has lattice points" to mean "the curve intersects a lattice point"
the last two sentences probably made the most sense out of all of that, my question basically boiled down to whether or not there's a formal notion of what I was calling a "local property"
@0x5f3759df well, I think the case of polynomials that map $\mathbb{Z}$ to $\mathbb{Z}$ is pretty well known, but I don't know if that is what you are looking for.
The underlying question I had was whether or not it's possible to make an oracle $L_b^a(f)$ which can tell you whether or not the function $f$ has a lattice point between $a$ and $b$
Everything I've come up with can't be reduced to a closed form, which is where I was originally getting my idea that this seems to be a "local property" that you can't even get general information about without evaluating the function at a specific point
@0x5f3759df if you're thinking about looking at "big things that happen" without paying as much attention to the "little things that happen" that could be a place to start
I don't know if my notation is canonical; this is abstract algebra, using the definition he gave us for $a\mid b$ in class, "a divides b", it would seem to not matter that we're talking about dividing by zero, merely that 0=0k?
Hey guys/gals, I asked a question on SE 7 days ago, and have edited it numerous times, and even though it has 3 upvotes more than downvotes, noone has answered it, how do I get more attention in general?
MSE I mean, and it should be a full stop before 'how do I...', sorry.
@Algebra is that the diagonal action one? have you made any progress or tried anything? it shouldn't take more than a few minutes, so I can only assume you don't really know what a group action is.
you have to check $ex=x$ and $g(hx)=(gh)x$ for all $x,g,h$
@anon If you look at the history of the question, I have my full attempt twice. Noone told me if it was right or wrong, so I deleted it fearing that people were scared by question size.
@Algebra If you fear that your question is too long, you can post your efforts in an answer. This was suggested on meta a few times, you can probably find some related discussion here or in the linked questions: Best way of asking “check my proof” questions
Is $\equiv$ the proper symbol to be using for the equivalence of the statements in this expression, or should I be using $=$. It is my understanding that the noninclusion of my bounds for a,b,c would not change the answer to this. If this is mistaken, I'm more fundamentally misled than I thought:
@MartinSleziak, these two together are my fear: http://meta.math.stackexchange.com/questions/13833/what-should-i-do-when-my-question-do-not-have-any-answer
@Algebra Ok, so it seems you have read the relevant meta threads. I don't think that I can give you any advice which wasn't already mentioned in one of them.
I don't think that your question is boring. And I appreciate that you made the effort. (Maybe you could post your attempts as an answer? Your question, you decide...)
I was thinking that might be the way to go. I have to say it would be strange to get three upvotes(likely for my efforts) and get no answers. Hopefully people will overview my answer, but -
"Another potential problem is that if the OP is primarily interested in the community's analysis of their own proof, it is less likely to get the community's attention, and thereby serve its purpose, if it's in an answer rather than in the original question." – Ben Blum-Smith, from the meta thread above mirrors my thoughts
@Algebra that webpage should have had a bunch of code before you clicked start chatjax... then after you click it, the code magically turns delicious into equations
I tried disabling them and it still doesn't work. Is the appropriate address for 'start chatjax' "http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML"
@Algebra It depends on how Facebook presents the text. On most static pages, "render MathJax" will work. However, dynamic, editable text may not always work with "start ChatJax" because of how that text is incorporated into the page. Also note that if Facebook is using https pages, ChatJax won't work.
