Hurkyl

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Feb 12	comment	Reference request - being rigorous about a common abuse of notation. @user18921: I'm probably being more complicated than I need to be (this is the first time I've thought this through). The point is, you can arrange all of your hypotheses, conventions, and assumptions into a diagram together with requirements on functors from that diagram to the topos. Limit and colimit cones can express many of those requirements: e.g. if the diagram has an object named "$A \times B$", I want a model to actually map that to a product of the interpretations of $A$ and of $B$, which is expressed as saying the functor must turn the limit cone into an actual limit.
Feb 12	comment	Ideals of $\mathbb{Z}[X]$ @QiL'8: Oh bother, you're right: height is an inf and I wanted a sup. :(
Feb 11	comment	What's the optimal strategy of this dice game? Your description of the game is very lacking. As described there are no choices a player can make: an entire game consists of one person rolling two dice a single time. I can imagine a variety of different games that it sounds like you were trying to describe. Also, you don't state what you are trying to find a strategy for. (common options are "maximal expected score", "best odds of beating some target score", and "best odds of beating an opponent", although the game as described is a single-player game....)
Feb 11	comment	When does a polynomial in $GF$ have a multiplicative inverse? @user61838: Polynomials are often used to represent elements of finite fields (in the same way that integers are often used to represent elements of the ring of integers modulo $n$). When you say "... in $\text{GF}(n)$", it sounds like you mean something that is an element of $\text{GF}(n)$. When "..." turns out to be "polynomial", on one hand it sounds like you really meant "... over $\text{GF}(n)$" , but on the other hand due to the aforementioned fact, it sounds like you really do mean elements of $\text{GF}(n)$, and that you're representing elements with polynomials. Thus ambiguity.
Feb 11	comment	Ideals of $\mathbb{Z}[X]$ @Martin: If $f \in I$, then there exists $q \in \mathbb{Q}$ and $h \in \mathbb{Q}[X]$ such that $fq = gh$. By adjusting $q$, we may assume $h \in \mathbb{Z}[X]$. If $g$ has content $1$, then $f = gh/q$ implies $g \| f$ (and that $h/q \in \mathbb{Z}[X]$).
Feb 11	comment	Evaluating $\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}$ If you set $y = 2^x$ and $z = 2^y$ and $f(x) = g(y) = h(z)$, the recursion "simplifies" to $$g(2y) = g(y)^2 - y \qquad \qquad h(z^2) = h(z)^2 - \log_2 z$$ I don't know either helps.
Feb 11	comment	When does a polynomial in $GF$ have a multiplicative inverse? Do you mean elements of the finite field $\text{GF}(n)$ or do you mean elements of the polynomial ring $\text{GF}(n)[X]$?
Feb 11	comment	Need help is defining an isomorphism. I assume $A$ isn't that matrix, but it's supposed to be the group of all such matrices? Anyways, why bother with a permutation group? Just find the isomorphism directly. The condition of which pairs of parameters to exclude should be a big hint as to the correspondence between them. It might help to write the elements of $A$ as pairs $(a,b)$ rather than as matrices, and work out the group law in terms of that representation.
Feb 11	comment	Ideals of $\mathbb{Z}[X]$ @Martin: If $I$ contains no integers, then we can invert every integer without killing $I$, making it an ideal of $\mathbb{Q}[X]$.
Feb 11	comment	Ideals of $\mathbb{Z}[X]$ Also, an ideal requiring $2$ generators may still satisfy $I \cap \mathbb{Z} = 0$, such as $I = (x^2, 2x)$
Feb 11	comment	Ideals of $\mathbb{Z}[X]$ The ideal $(2,x)^n$ requires $n+1$ generators.
Feb 11	comment	A question on polynomial If one is interested in exact statements, the right hand side actually has at most 6 poles, because both terms share the common factor $(t-1)$ in the denominator.
Feb 11	comment	Is infinity a number? For the sake of precision, I want to point out that @Ben is interpreting "$\infty$" in non-standard analysis to denote an infinite (nonstandard) number, rather than to denote to the transfer of a standard use of the symbol $\infty$.
Feb 11	comment	Choosing $n$ such that $(n/p) = -1$ for all $p$ @Gerry: I'm not sure either.
Feb 11	comment	Does the concept of localizing at an extension of a prime ideal make sense? My description of nice situations is rather narrow -- I wanted to make sure I was saying something true, but I'm sure something similar can be said in great generality.
Feb 11	comment	Reference request - being rigorous about a common abuse of notation. sketch at nLab and sketch at wikipedia
Feb 11	comment	Reference request - being rigorous about a common abuse of notation. $\mathbb{N}$ and $\mathbb{R}$ aren't really objects of the topos: they're objects of a sketch. The chosen canonical arrow is taken from the sketch, and thus yields an appropriate arrow for every model of the sketch in the topos.
Feb 10	comment	Is $\sqrt 7$ the sum of roots of unity? @MJD: Yes, because the ring of integers in $\mathbb{Q}(\zeta)$ is $\mathbb{Z}[\zeta]$, and $\sqrt{d}$ is an algebraic integer. (I don't follow your recipe)
Feb 10	comment	A improper integral on expontential Have you tried just multiplying it out?
Feb 10	comment	Question about algebraic closures @Qiaochu: To add rigor while maintaining the spirit of the question, it might be better to phrase it as asking if the field extension $\overline{K}/\mathbb{Q}$ is isomorphic to $\overline{\mathbb{Q}}/\mathbb{Q}$.