TITLE: How to show the existence of an ideal in the ring of Gaussian integers that satisfy the following?
QUESTION [2 upvotes]: How to show that

If $p$ is a prime and $p\equiv1\bmod4$ then there exists an ideal of the $R=\mathbb{Z}[i]$, the ring of Gaussian integers, such that $R/I$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$.

I don't know how to approach this problem. These are some things I noted while scribbling.
If we argue that the canonical homomorphism '$f$' from $R$ to $R/I$ takes $i$ to $f(i)$ such that  $f(i)^2 = -1 \bmod p$ because $i^2 = - 1 \bmod p$.
The element $(p-1)/2 !$ in $\mathbb{Z}/p\mathbb{Z}$ satisfies this. Hence,
$f(i) = (p-1)/2 !$, because using Wilson's Theorem that  $(p-1)! = -1 \bmod p$ gives us  $( (p-1)/2 ! )^2 = -1 \bmod p$.
I will really appreciate some help on this. Thank you so much.

REPLY [2 votes]: You found an $a\in\mathbb Z$ such that $p\mid a^2+1$. If $p$ is prime in $\mathbb Z[i]$ then, from $p\mid a^2+1$ you get $p\mid a+i$ or $p\mid a-i$, and both cases lead to a contradiction. This shows that $p$ isn't prime in $\mathbb Z[i]$. Then $p=(m+ni)(m-ni)$, so $p=m^2+n^2$. Now just take the ideal generated by $m+ni$.


TITLE: Are these functions ordered correctly from smallest to largest growth rate?
QUESTION [0 upvotes]: I've attempted to put a random list of functions into order from smallest to largest growth rate, but I seem to be doubting myself quite a bit. Does this look accurate? If not, please feel free to point out my errors.
$$
128, 2/N, \sqrt N, 23N, N \log_2 N, N^2 \log_2 N, 42N^3, 2^N, 2^{N+1}, 3^N, N!
$$
Thank you!

REPLY [1 votes]: $$2/n,128,\sqrt n,n\log_2n,n^2\log_2n,$$
and the rest is correct. I'm assuming you mean growth as $n\to\infty$.


TITLE: What is the probability that a graph containing n nodes with k random edges each will be strongly connected?
QUESTION [3 upvotes]: Consider a "random" undirected graph, with n nodes and (on average) k edges assigned to each node, such that the edge connects the node to a randomly chosen node in the graph. What is the probability that this graph will be connected, i.e. contain a path from every node to every other node?

REPLY [1 votes]: See wikipedia about the Erdos-Renyi model. Two key properties are:

If $pn<(1-\epsilon)\log n$, then the graph is almost surely disconnected.
If $pn>(1+\epsilon)\log n$, then the graph is almost surely connected.