Eh, I don't think of ideals as numbers. Rather, individual elements of a ring can be sort of "avatars" for ideals, but this only works in PIDs. @Skull: The examples I gave might take some algebraic sophistication to fully understand. (I don't really know much about id/adeles for example.)
Depends on what kind of infinity. There's R or C with a "point at infinity," but this is more topological than algebraic, and cardinal arithmetic is certainly a good example of borderline numbers.
Coming a little late to the conversation, but I don't think you'd find many people, if any, who'd call ideals "numbers." In fact, in a pretty literal sense they were originally introduced to tidy up Kummer's notion of "ideal numbers," which were very tedious to work with.
I think id/adeles make for a good candidate for a borderline case.
An ideal $M$ of a commutative ring $A$ (with unity) is maximal iff $A/M$ is a field.
This is easy with the correspondence of ideals of $A/I$ with ideals of $A$ containing $I$, but how can you prove it directly? Take $x + M \in A/M$. How can you construct $y + M \in A/M$ such that $xy - 1 \in M...
So if you're picking the {F_n} as I assume you are, then that converges uniformly to F. That's good. So, as you said, we need to calculate the integral of xF_n(x) over that interval.
@BenjaminLim Mostly it's getting used the language of exact sequences. And those just organize information about kernels, images, and cokernels that you already know.
@DevenWare So suppose I know $\int_0^1 xF_n(x) dx$. Look at $\int_0^{1/3} xF_{n + 1}(x) dx$, for example. Write $u = 3x$ and work out the details; you get $\frac{1}{18}\int_0^1 uF_n(u) du$.
1/2 just comes from the 1/2 in the formula. Another 1/3 from the u, another 1/3 from the du.
They don't recall. They define what a ring is so I assume that they mention the definition at some point.
But it's just a definition.
I think there's some tacit assumption that you've seen rings before, though. I think if you introduced someone to ideals for the first time using AM it would not go well.
I guess the hope is that you've seen those words before. Certainly when talking about polynomial rings or matrix rings the notion of an $A$-algebra is natural.
@Dylan Sorry I think maybe i'm missing something silly how did you get to $F_n$ instead of $F_{n+1}$? "Write $u = 3x$ and work out the details; you get $\frac{1}{18}\int_0^1 uF_n(u) du$."
@DylanMoreland Not positively, but not especially negatively, either.
@DylanMoreland I’ve never seen either of his books, but his standard graduate algebra course at Madison was superb. His lectures were so well organized that my notes very nearly are a book.
@DylanMoreland I wasn’t even thinking along those lines $-$ just noting that I seem to be less strongly affected by this type of question than a lot of folks are.
I think that it’s more a matter of some 40 years’ practice and exposure to students’s questions: I tend to read those questions as the result of some combination of desperation and general cluelessness. There are certainly other things that can get me worked up quickly enough!
Okay Dylan sorry I'm quite slow at integration and concepts like this just getting to understand it, so we've got $\frac{1}{18}\int_0^1 uF_n(u) du for \int^{1/3}_0 F_{n+1} and we can do \int^{1}_{2/3} similarly, and $\int^{2/3}_{1/3} is always the 1/12 correct? but then I still can't see how to conclude
just take the limit as n\rightarrow \infty of the sum of these three values?
but the ones on the edges seem to go to 0 unless I'm seeing it wrong, but that seems wrong to me, e.g it seems wrong that in 0,1/3 and 2/3,1 the integral would be 0
I suppose not, however at the point of going down and figuring that out the solution seems to be getting a bit more convoluted than I would expect it to be
In the end, I think you want some sequence of integrals $I_n \to I$, and you have some relation $I_{n + 1} = aI_{n} + b$, hopefully. And you'd take the limits there and find $I$.
I agree that this seems messy.
I didn't really follow your TA's hint, unfortunately.
I mean, I'm just going to write down what he said, I bet: for $x \in [0, 1/3]$, $F(x) = \frac{1}{2}F(3x)$, and so on. No need for the $F_n$s. This makes sense from the picture and the finite approximations. Maybe it's clearest if you define the function using the ternary expansion.
I think that once you write all of this out and use that fact about $\int_0^1 F$, then you'll write $I = \int_0^1 xF(x)\, dx$ in terms of itself, in the sense that you'll have
$I = a \cdot I + b$ for some numbers that you'll have to work out. I'm almost to the end but I should stop. And then you can solve for $I$.
@VVV Do you know that $f \in F[x]$ with $deg(f) = 2$ or $deg(f) = 3$ is reducible over $F$ if and only if it has a zero in $F$? So you just pick a degree $2$ polynomial that doesn't have a zero, like for example $x^2 - x + 1$.
@AsafKaragila that's why I'm asking. He seem to be surprised by the version of Galois' death I've told him yesterday. Even the fact that it was extracted from the notes by Euclid didn't convince him: he proceed asking such respectable members as Rob and Brian if it is true
@MattN it's already warmer here. I hope that winter will finish this week
@Asaf: to you, my friend, the winter is the time of the year when it is colder than $0^\circ \;C$ and there is some... snow instead of the rain. Snow is like rain but instead of drops of water you have milidrops of ice. And ice is the state of the water for temperatures below $0^\circ \;C$
@Ilya Suppose that $U$ is an open subset of $X\times Y$ and $x_0\in X$. Let $U_{x_0}=\{y\in Y:\langle x,y\rangle\in U\}$, and suppose that $y\in U_{x_0}$. Then $\langle x_0,y\rangle\in U$, so there are open nbhds $V$ and $W$ of $x_0$ and $y$, resp. such that $V\times W\subseteq U$. Clearly $y\in W\subseteq U_{x_0}$.
For me dinner means the main meal of the day, whenever it comes. So I might have dinner and supper, or I might have lunch and dinner. (Except that I don’t usually eat three meals a day.)
@BrianMScott do you usually use the word 'supper' at all in US? I ask you because on our English classes we were told that classical 3-meals sequence is breakfast(morning)-dinner(midday)-supper(evening), and then it appeared that it is breakfast(morning)-lunch(midday)-dinner(evening)
People who follow the B-D-S pattern tend to have dinner in the afternoon and supper quite late; people who follow the B-L-D pattern often have lunch around noon and may have dinner as early as 17:00.
I think that in the U.S. the B-L-D pattern is by far the more common. I associate the other pattern with those who don’t need to work for a living.
Yes: when she was young, my mother would insult a classmate by calling him a one-seeded fruit with a woody pericarp; this was apparently a dictionary definition of nut.
@Ilya My knowledge of obscene Russian is almost non-existent, though I do know the verb that descends from Proto-Indo-European $^*h_2yeb^h$-; you can probably guess it if you ignore the $h_2$.
I have a problem, " $S$ is the set of integers in the form $a^2+4ab+b^2$ where $a$ and $b$ are integers. If $x$ and $y$ are members of S how can you prove that $xy$ is also in S?"
An ideal $M$ of a commutative ring $A$ (with unity) is maximal iff $A/M$ is a field.
This is easy with the correspondence of ideals of $A/I$ with ideals of $A$ containing $I$, but how can you prove it directly? Take $x + M \in A/M$. How can you construct $y + M \in A/M$ such that $xy - 1 \in M...
