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2d

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Fermat's last theorem over larger fields
Could a Russian speaker add a summary of the results in this paper to this answer, please ? 
2d

answered  On a minimal algebraic number field which satisfies the principal ideal theorem 
Nov 22 
awarded  Nice Question 
Nov 20 
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Is elliptic curve point division defined over the field of real numbers?
If you are looking for "procedures" to compute it for general fields, then look at sage's function P.division_points(n). It starts by finding roots of the division polynomial in the field. 
Nov 20 
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elliptic curves and tower of finite fields
What do you mean by "reduce"? To me it seems only when $m$ divides $n$, there is a natural map, the trace map, which sums the Galois conjugate points. 
Nov 20 
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How to explicitly compute lifting of points from an elliptic curve to a modular curve?
If increasing the conductor is computationally hard, maybe you could enlarge the base field to get to rank 2 examples. Find a small conductor curve with large rank over a small quadratic field. 
Nov 17 
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Does this equation has a closedform solution for $t$? ($(1p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1t)^i)$)
The Galois group of the equation for $p=3/5$ and $n=7$ is $S_7$, which is not soluble. So don't expect a solution by radicals for $n=7$. 
Nov 12 
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For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?
Yes that is possible for $p=5$. It turns out to be equivalent to be a quadratic twist by 5 of a curve with a rational 5torsion point. 
Nov 11 
revised 
For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?
added 108 characters in body 
Nov 11 
comment 
For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?
You are right. More generally, If $G$ is the group of all matrices of the form $(\begin{smallmatrix} 1 & * \\ 0 & * \end{smallmatrix})$, then $H^1(G,E[p])= \mathbb{F}_p$ if $p =3$ and it is zero if $p>3$. Your example, and many other curves with a 3torsion point rational over $\mathbb{Q}$, has indeed this group. 
Nov 11 
revised 
For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?
deleted 4 characters in body 
Nov 11 
answered  For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial? 
Nov 11 
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For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?
I don't know of an example with $p>2$. Maybe the group theorists can tell us examples of $H^1(H, V)\neq 0$ for subgroups $H$ of $\operatorname{SL}_2(\mathbb{F}_p)$ with $V$ the 2dimensional vector space over $\mathbb{F}_p$. Say with $p=3$ or $p=5$. 
Nov 11 
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For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?
Over $\mathbb{Q}$, the determinant $G\to \mathbb{F}_p^{\times}$ must be surjective. So your $G$ won't appear as a group for an elliptic curve over $\mathbb{Q}$. 
Nov 9 
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A number theoretic identity
it is true at least up to $2n+1<1000$. 
Nov 9 
awarded  Enlightened 
Nov 9 
awarded  Nice Answer 
Oct 24 
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$j$invariants of elliptic curves over finite fields
For the second, look up "twists" in chapter X of Silverman's book. Together with the knowledge of what the endomorphism ring is yo ushould get the answer. 
Oct 16 
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Is the CasselsTate pairing defined for elliptic curves over function fields?
There are though known problems with the proof that the kernels are the maximal divisible subgroup. See HarariSzamuely and GonzalezAviles for the corrections, both in Crelle 632 in 2009 with a title containing "arithmetic duality theorems". 
Sep 19 
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What are all positive integers n for which the congruence $a^{n+1} \equiv a (mod n)$ holds?
Moreover this is a duplicate from mathoverflow.net/questions/37097/… 