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Four is possible, an example is $0,451584,462400,485809$. This solution comes from the example of a cuboid with integer sides, space diagonal and two out of three face diagonals given here. I don't kn …

No. If $K$ is Galois, since the Galois group act transitively on the $\mathfrak{p_i}$, their class must be the same up to automorphism of the class group. Thus you can take as a counterexample any Gal …

You a looking for the number of $k$-potent elements of the ring $\mathbb{Z}[i]/(n)$. If $n=p_1^{r_1}\cdots p_s^{r_s}$ is the prime factorization of $n$, then by the Chinese Remainder Theorem the ring …

One way to formulate local class field theory is by saying that the local Artin map induces an isomorphism from the profinite completion of $K^\times$ to $\operatorname{Gal}(K^\text{ab}/K)$, which sat …

In Discrete groups, expanding graphs, and invariant measures (in the notes at the end of Chapter 7), Lubotzky mentions that certain estimates for the number of spanning trees $\kappa(G)$ of a $k$-regu …

The condition for a connected $(q+1)$-regular graph to be Ramanujan is that every nonzero eigenvalue $\lambda$ of the graph Laplacian satisfy $$q+1-2\sqrt{q}\le \lambda\le q+1+2\sqrt{q}.$$ With a litt …

Here is a simple proof in the case when $K$ has characteristic $2$. Let $m = \frac{n}{2}$. For $0\le i < m$, and $x\in K^n$, let $p_i(x)=(x_{2i},x_{2i+1})$. I claim that for any fixed $0\le i < m$, $p …

Dimension 4 is clearly impossible since your quadratic forms are isotropic, but dimension 3 is possible. The following SageMath code generate random 3-dimensional subspaces and check whether they are …