34 Calculate $\int_{0}^{1} (x-f(x))^{2016} dx$, given $f(f(x))=x$. 32 Are there infinitely many primes $p$ such that $p-2$ and $p+2$ are composite? 24 The square roots of different primes are linearly independent over the field of rationals 22 Elementary solution of exponential Diophantine equation $2^x - 3^y = 7$. 16 Why does “solvability” for groups suggest something about the solution of polynomials?

### Reputation (16,016)

 +10 Prove that a finitely-generated free module has a finite basis +65 A polynomial divisible by primes of special form +10 Calculate $\int_{0}^{1} (x-f(x))^{2016} dx$, given $f(f(x))=x$. +15 If $R=\mathbb{Z}[\sqrt{p}]$ with $p$ an odd prime and $I=(2,1+\sqrt{p})$, then $I^2=(2)$

### Questions (6)

 7 A reference/proof request for a result on real radical extensions 6 Is there an algebraic extension $F \neq \bar{\mathbf Q}$ of $\mathbf Q$ such that every irreducible polynomial in $F[X]$ has odd degree? 4 The irreducibility of $X^q - 3$ in the field extension $\mathbb{Q}(\zeta_q, 2^{1/q})$ 3 Embedding the absolute Galois group of $\mathbf Q_p$ into that of $\mathbf Q$ 3 Is my proof that $\mathbb{Q}(\sqrt{5})$ is the only quadratic subfield of $\mathbb{Q}(\zeta_5)$ correct? [duplicate]

### Tags (222)

 398 abstract-algebra × 142 126 group-theory × 42 241 field-theory × 81 119 extension-field × 44 213 number-theory × 53 118 elementary-number-theory × 30 193 galois-theory × 67 102 linear-algebra × 48 130 algebraic-number-theory × 53 101 prime-numbers × 11

### Bookmarks (30)

 109 Construction of a Borel set with positive but not full measure in each interval 81 If polynomials are almost surjective over a field, is the field algebraically closed? 80 How to use the Extended Euclidean Algorithm manually? 59 Simplicial Complex vs Delta Complex vs CW Complex 33 If $(A-B)^2=AB$, prove that $\det(AB-BA)=0$.