All Questions
Tagged with galois-theory prime-numbers
7 questions
3
votes
1
answer
188
views
Is there a uniform family of polynomials $f_p(x) =x^2 + a(p)x + b(p)$ such that $f_p(x)\in \mathbb{Z}[x]$ is irreducible and irreducible mod $p$?
Let $p\in\mathbb{Z}$ be a positive prime number.
Is there a "uniform" family of polynomials $f_p(x) =x^2 + a(p)x + b(p)$ of degree two such that $f_p(x)\in \mathbb{Z}[x]$ is both irreducible ...
- 237
2
votes
0
answers
147
views
Prime splitting in the division field of an elliptic curve
Let $E/\mathbb{Q}$ be an elliptic curve with good reduction at two distinct primes $p, \ell$. Suppose the mod $\ell$ Galois representation associated to $E$ is surjective. Let $K=\mathbb{Q}(E[\ell])$ ...
- 1,422
1
vote
1
answer
124
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why is $R := \mathbb{Z}p +XF[[X]]$ an almost valuation domain?
Recall that an integral domain $R$ with quotient field $K$ is
an almost valuation domain if for every $0 \not= x \in K$, there is a positive integer
$n$ (depending on $x$) such that $x^n \in R$ or $x^{...
- 147
10
votes
2
answers
1k
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Is a Galois extension of the rationals determined by its set of completely split primes?
apologies if this is a naive question. Consider two Galois extensions, K and L, of the rational numbers. For each extension, consider the set of rational primes that split completely in the extensions,...
- 595
12
votes
1
answer
1k
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Least prime $p$ such that an irreducible polynomial of degree $n$ has no root modulo $p$?
This question is inspired by an old question of Greg Kuperberg, about how small is the first prime $p$ which makes a given monic polynomial $P$ with integral coefficient have a (simple) root modulo $p$...
- 26k
1
vote
1
answer
1k
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What does Gal(Q_p/Q) mean? [closed]
What does
$\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q})$ mean? ($p$ is a prime number.)
If it is defined as $\mathrm{Aut}(\overline{\mathbb{Q}}_{p}/\mathbb{Q})$, then does it have any property ...
- 945
11
votes
1
answer
2k
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Maximal extension almost everywhere unramified and totally split at one place
Fix a finite set of primes $S$ and an additional prime $p$. Let $K$ be the maximal extension of $\mathbb{Q}$ that is unramified outside $S$ and $\infty$ and totally split at $p$. Is the extension $K$ ...
- 21.3k