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Taking the question in the title literally this is quite easy. Here's how to take a (finite order) Tate character and derive the corresponding Dirichlet character (omitting most details). Let $\chi:I …

Yes, this follows from the analytic class number formula. See http://www.math.niu.edu/~rusin/known-math/97/sign . Added I have now found a reference. This is a theorem of Mordell: L. J. Mordell, Th …

One way (I don't know whether this is done in practice) is to go via the intermediate quadratic extension. Let $L$ be the $S_3$ extension. Then it has a quadratic subfield $K$. There is a relation bet …

Well you can lift any set of generators for the finite group $U/U^p$ and they will be a set of generators of $U$. If $n$ is the degree of $L$ over $\mathbb{Q}_p$ then $U/U^p$ will have order $p^{n+1}$ …

It's well-known that if $a$ is an integer then a prime factor of the number $\Phi_n(a)$ is either a factor of $n$ or congruent to $1$ modulo $n$. Here $\Phi_n$ is the $n$-th cyclotomic polynomial. The …

How about $x+2\cos(2\pi/7)y+4\cos^2(2\pi/7)z$, $x+2\cos(4\pi/7)y+4\cos^2(4\pi/7)z$, $x+2\cos(6\pi/7)y+4\cos^2(6\pi/7)z$ ?

As Daniel has pointed out, there is an elementary proof that for each $n$ there are infinitely many primes $p$ with $p\equiv1 \pmod n$. There is an also an elementary proof that for each $n$ there are …

An interesting example where the resultant and the "reduced resultant" differ comes from the theory of elliptic curves. Take an elliptic curve $$E:\qquad y^2=x^3+ax+b$$ where $a$ and $b$ are integers. …

You might consult the following paper John Brillhart & Patrick Morton 'Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial', Journal of Number The …

Your equation $n^2=(b+1)(b^2+1)$ defines an elliptic curve. By Siegel's theorem http://en.wikipedia.org/wiki/Siegel%27s_theorem_on_integral_points the set of integer solutions will be finite. Mordell' …

I'll write $\chi(x)$ for the Legendre symbol modulo $p$. Consider $$f(x)=(\chi(x)+1)(\chi(x-1)-1)(\chi(x+1)-1).$$ Then $f(x)=8$ if $x$ is an isolated quadratic residue and $0$ otherwise (unless $x$ i …

According to the notes in fifth edition of Niven, Zuckerman and Montgomery's An Introduction to the Theory of Numbers for each $\epsilon\in(0,1)$ there is a series of parameters in Chebyshev's method …

Accoring to H. Lenstra, Chebotarev's theorem holds both for Dirichlet and for natural density (but he doesn't give a reference in this document). Applying Chebotarev to the extension $H/\mathbb{Q}$ wh …

There was a great deal of discussion in the sci.math newsgroup about a decade ago. See the threads Beal's Conjecture and Against the term "Beal Conjecture". As with most sci.math discussions, they gen …

I'm not sure what you mean by a random integer $n$, but would you agree that the probability that a random squarefree integer be divisible by $55$ is nonzero? For if $55\mid n$ then $5\mid\gcd(n,\phi( …