1. ## Algebra

a) find the degree of a minimal splitting field of (x^6 + 1) over Q.

b) if a is a any algebraic number prove that there exists a positive integer 'n' such that 'na' is an algebraic number.

c) if W is a subspace of V and vєV satisfies (v,w) + (w,v) = (w,w) for all wєW prove that (v, w)=0 for all wєW where V is inner product space over F.

d) if T1,T2 є Hom(V,W), then show that
1) r(α(T1))=r(T1) for all αєF, α≠0
2) | r(T1)-r(T2) | ≤ r(T1+T2) ≤ r(T1) +r(T2), where r(T) means rank of T.

2. Originally Posted by mslghlg

so many questions and not even one line work from you!

a) find the degree of a minimal splitting field of (x^6 + 1) over Q.
find the roots of $\displaystyle x^6=-1$ and then find the smallest field which contains the roots and $\displaystyle \mathbb{Q}.$ your final answer is $\displaystyle \mathbb{Q}(i,\sqrt{3}).$

b) if a is a any algebraic number prove that there exists a positive integer 'n' such that 'na' is an algebraic number.
are you sure they don't want you to prove that $\displaystyle na$ is an algebraic integer not just an algebraic number, which trivially is if you let n = 1.

c) if W is a subspace of V and vєV satisfies (v,w) + (w,v) = (w,w) for all wєW prove that (v, w)=0 for all wєW where V is inner product space over F.
is the field F here supposed to be the field of real or complex numbers? in either case noting that $\displaystyle (w,w) \geq 0$ and changing w to -w in your identity will solve the problem.

3. "if a is a any algebraic number prove that there exists a positive integer 'n' such that 'na' is an algebraic integer"