4,815 reputation
1729
bio website Workingonit...
location Pasadena, CA
age 19
visits member for 1 year, 7 months
seen 2 hours ago

Undergraduate freshman at Caltech. PROMYS student and counselor.

Things I am confused about, in the form of media titles:

  • The Trouble with Tensors
  • My Little Functor: Category Theory is Magic

Aug
29
comment Calculate quotients of $\mathbb S_3$ and $\mathbb D_4$
A lot of people are giving answers, but here's a way of actually, you know, doing the problem. You know how you build subgroups, right? Pick an element, then multiply it by everything in the group until it "closes up". To build normal subgroups, do the same thing, but every time you add a new element, add all its conjugates as well. All the tricks for "wait, did I make this subgroup already" will still apply.
Aug
27
comment How to show that set of strings of odd length in $\{a,b,c\}^*$ is countable?
Diagonalization is typically used to show something isn't countable. For this problem, use the fact that "the union of countably many finite sets is also countable".
Aug
20
answered Adjacent non-coprime numbers
Aug
19
revised Modular arithmetic and linear congruences
added 48 characters in body
Aug
19
comment Modular arithmetic and linear congruences
You're trying to solve $ax \equiv b \pmod m$, not $x \equiv ba^{-1}$. The latter only makes sense when $\gcd(a,m) = 1$. But the former always makes sense. Also, "If there is no inverse, there is no solution." is just false. Look at my previous comments.
Aug
19
comment Modular arithmetic and linear congruences
But such an equation can have solutions. Let $a = 2$ and $b = 4$. You can't invert $2$ in the integers, but there's definitely a solution. (Admittedly, this is a bit unsatisfying as a response, because we can just pass to $\mathbb{Q}$, and find an inverse there. I'll have to think of a better example. How much do you like matrices?)
Aug
19
comment Modular arithmetic and linear congruences
Are you asking "how can it have a solution" or "how can we find the solutions"? I answered the latter above, but as for the former: The point is that you don't need an inverse. For example, it's pretty easy to find a solution to $2x \equiv 4 \pmod {10}$, even though $2$ and $10$ are not coprime (note that there's another solution though: $7$). This is one of those intuitions that can easily be built by trying lots of examples.
Aug
19
answered Modular arithmetic and linear congruences
Aug
19
answered What happens if the coefficients of polynomials are not taken from a field of real numbers?
Aug
17
answered How to find a modular multiplicative inverse when GCD is not 1
Aug
14
revised Field theorem: impossible to satisfy three equations simultaneously in an integer field?
added 11 characters in body
Aug
14
answered Field theorem: impossible to satisfy three equations simultaneously in an integer field?
Aug
12
comment Inclusion of Fields whose order is a prime power
Do you know how the polynomial $x^{p^k} - x$ behaves in $\mathbb{F}_{p^k}$?
Aug
7
answered What exactly is a maximal ideal?
Aug
4
revised Primes and the number of digits of the prime
This user is posting PROMYS questions verbatim, and I've been asked to edit them so exact copies of the problem set aren't available online.
Aug
4
revised Number of roots of polynomials in $\mathbb Z/p \mathbb Z [x]$
This user is posting PROMYS questions verbatim, and I've been asked to edit them so exact copies of the problem set aren't available online.
Aug
4
revised number theory proofs with units, orders, and the phi function
This user is posting PROMYS questions verbatim, and I've been asked to edit them so exact copies of the problem set aren't available online.
Aug
4
revised geometric methods in number theory
This user is posting PROMYS questions verbatim, and I've been asked to edit them so exact copies of the problem set aren't available online.
Aug
4
revised Finding roots of equation in $\Bbb{Z}_{14}$, $\Bbb{Z}_{17}$
deleted 94 characters in body
Aug
4
revised How to find the roots of polynomials in $\Bbb Z_p$
This user is posting PROMYS questions verbatim, and I've been asked to edit them so exact copies of the problem set aren't available online.