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We know that in a normal extension of number fields $L/K$, for any prime $P$ of $K$ and any primes $Q_1,Q_2$ of $L$ lying over $P$, the ramification indices and inertial degrees are the same, $$e(Q_1| …

Here is something I noticed - no idea if it helps: Suppose $A\subset\mathbb{Z}$ has the property that for each odd prime $p$ and $\phi_p:\mathbb{Z}\rightarrow\mathbb{Z}/p\mathbb{Z}$, we have $|\phi_p …

In the remark below the statement of Euler's criterion in the paper you linked to, notice that Grube, who tried to correct Euler's original "proof" of this criterion, actually only provided a correct …

Your proof looks right to me. As a smaller example, a number with a Zeckendorf representation with 7 terms is 609 = 377 + 144 + 55 + 21 + 8 + 3 + 1 (note that 609 = 610 - 1); I believe it is impossibl …

My 2 cents (ha ha): perhaps the approach is to show that the power of 2 dividing $2+\sum_{i=1}^{n-1}a_i^2$ is eventually less than the power of 2 in $n$, and that this somehow involves looking at $a_{ …

The answer is no. Let $\alpha\in\mathbb{C}$ be a root of $x^3+tx+1$ for $t\in\mathbb{Q}\setminus\mathbb{Z}$, and let $L=\mathbb{Q}(\alpha)$, so that $N_\mathbb{Q}^L(\alpha)=1$ and $tr_{\mathbb{Q}}^L(\ …

This seems like an obvious fact, but I'm not sure what the necessary meaning of "nice" is to get a result like this. I'm wondering if there is a theorem of the form: For any <1> field extension $K/F …

The standard definition is that $a\in\mathbb{Z}$ is a primitive root modulo $p$ if the order of $a$ modulo $p$ is $p-1$. Let me rephrase, to motivate my generalization: $a\in\mathbb{Z}$ is a primitiv …

The conflict is just that some people use the words valuation and absolute value interchangeably. The term "p-adic valuation", used correctly, refers to $\nu$, though perhaps in some areas of math the …

Yes, there is such a thing as the irrationality measure of a real number (I'm not sure if it can be / has already been extended to complex numbers). It is based on the idea that all algebraic numbers …

I think Jones + Jones http://www.amazon.com/Elementary-Number-Theory-Gareth-Jones/dp/3540761977 would be a good all-around introduction. It has solutions to every problem in the back, which can be …

So, equivalently, suppose we have a symmetric function $S( , , , )$ such that $S(z_1,z_2,z_3,z_4)=0$ whenever $z_1,z_2,z_3,z_4$ are conjugates over $\mathbb{Q}$ and such that $\mathbb{Q}(z_i)\supset\m …

It depends on what you mean by "calculate". If you want a "closed-form" expression for $k(m)$, the period of the Fibonacci sequence modulo $m$: One should expect that there is no such thing, because …

Why exactly is the unique factorization of elements into irreducibles a natural thing to look for? Of course, it's true in $\mathbb{Z}$ and we'd like to see where else it is true; also, regardless of …

If we can make a (single variable) polynomial function $g(x)$ from $\mathbb{Z}$ onto $\mathbb{N}$, we could compose it with the Cantor pairing function, but such a $g(x)$ seems implausible for some re …