4 Continuity of a function $\varphi : \ell_1 \to \mathbb{R}$ defined by $\varphi \left (\left ( x_n \right )_{n=1}^{\infty} \right )=\sum x_n^2$ 3 prove that every Sylow p-subgroup of $S_n$ is abelian if and only if $n ### Reputation (289)  +5 Find the intersection of all$T_2$topologies on an infinite set$X$+5 If$k_1, \ldots, k_n$are non-square, pairwise coprime, then$\sqrt {k_n} \not \in \mathbf{Q}(\sqrt {k_1}, \ldots, \sqrt {k_{n-1}})$+10 With$[K: F] = 3, \alpha \in K, \beta \in K \backslash F$, show that there are$a,b,c,d \in F$such that$\alpha = \frac{a+b\beta}{c+d\beta}$+5 prove that every Sylow p-subgroup of$S_n$is abelian if and only if$n

 1 Is maths inductive or deductive? 0 With $[K: F] = 3, \alpha \in K, \beta \in K \backslash F$, show that there are $a,b,c,d \in F$ such that $\alpha = \frac{a+b\beta}{c+d\beta}$

### Tags (35)

 1 logic 0 graph-theory × 3 0 general-topology × 6 0 group-theory × 3 0 functional-analysis × 5 0 hilbert-spaces × 3 0 extension-field × 3 0 ordinary-differential-equations × 3 0 field-theory × 3 0 measure-theory × 2