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13 votes

How to prove that the sum and product of two algebraic numbers is algebraic?

abstract-algebra field-theory asked Jun 7 '12 at 13:42
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13 votes

Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$

abstract-algebra polynomials field-theory galois-theory finite-fields asked Jun 20 '12 at 16:24
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11 votes

Is there a non-cyclic group with every subgroup characteristic?

abstract-algebra group-theory asked May 29 '12 at 19:58
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10 votes

Is there an infinite simple group with no element of order $2$?

group-theory finite-groups simple-groups asked May 1 '12 at 22:56
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10 votes

For each $n$, does there exist a prime $p$ and integer $k$ such that $p^k - 1$ has exactly $n$ prime divisors?

elementary-number-theory prime-numbers asked Feb 1 at 16:14
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7 votes

Does $\varphi(mn) = \varphi(m)\varphi(n)$ imply that $\gcd(m,n) = 1$?

elementary-number-theory asked Feb 26 '12 at 19:33
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7 votes

Every group of order $p^5$ is metabelian

group-theory finite-groups asked Mar 24 '12 at 18:51
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7 votes

Is there a nontrivial semidirect product of two groups isomorphic to their direct product?

abstract-algebra group-theory asked Sep 24 '12 at 15:05
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7 votes

Is the asymptotic density of powers of primes zero?

elementary-number-theory asked Jan 27 at 23:08
math.stackexchange.com

6 votes

If $f(x) \in F[x]$ is a polynomial solvable by radicals, does this imply that its Galois group is solvable when $F$ has characteristic $p > 0$?

abstract-algebra galois-theory asked Jun 24 '12 at 18:52
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