| Front | [Conditions for separability] Let [$]f\in K[x][/$] be irred. Then |
| Back | (i) f separable iff [$]f'\neq 0[/$](ii) if charK=0 then f is separable(ii)if char K=p, then f inseparable iff [$]f(x)=g(x^p)[/$] for some gProof:(i) gcd(f,f')= 1 or f as f is irreducible.If [$]f'\neq 0[/$] then by looking at the degree [$]gcd(f,f')=1[/$] so separable.Other case is easier(ii) and (iii) [$]f'(x)=0[/$] iff [$]ir_i=0[/$] for all [$]i\geq 1[/$]in characteristic 0 that's true only for constantsin characteristic p that also gives us what we want |
| Tags | separable |
| Front | [Proposition 3.6.4.] Let Eˆ = E2 ⊗ E1 and µˆ = µ2 ⊗ µ1. For a function f on E1 × E2, write ˆf for the function on E2 × E1 given by ˆf(x2, x1) = f(x1, x2). Let f be a nonnegative E-measurable function. Then |
| Back | ˆf is a non-negative Eˆ-measurable function and µˆ( ˆf) = µ(f). |
| Tags | product_meas |
| Front | [Solvable group definition] G is solvable is there is a chain of subgroups |
| Back | [$]1=G_0\leq G_1\leq ...\leq G_r=G[/$] with (i) [$]G_i\unlhd G_{i+1}[/$](ii) [$]G_{i+1}/G_i[/$] is abelian(ii)' [$]G_{i+1}/G_i[/$] is cyclic |
| Tags | solvable |