Galous Group Modules Self Study: Q&A

[tohauz]

I was doing some self study and have questions:
1. p(x)=x^{7}+11 over Q(a), R.
where a is 7-th root of unity. What are Galouis groups?
For the 1st case I got Z_{7}, second not sure. need hint for that
2. need hint. I know it is easy: M is an R-module. Show that Hom_{R}(R,M)$$\cong$$M.
3. Spse that I is an ideal of R such that I^{k}=0 for some k>0 integer. Let M, N be R-modules and let $$\phi$$:M->N be an R-module hom. Prove that if induced map $$\bar{\phi}$$:M/IM->N/IN is surjective, then $$\phi$$ is surjective.
4. show that 2$$\otimes$$1 $$\neq$$0 in 2Z$$\otimes$$Z/2Z.

 

[matt grime]

2. You need a map from

{f : R-->M } to M

(or vice versa).Assuming that R is unital, then there is only one possible map you can write down:

f ---> f(1)

You have to try to show that is an isomorphism. You also may want to think about the other direction;

M --> {f: R --> M}

again, there is only one possible map you can write down - given m in M, then the only candidate in Hom_R(R,M) is translation by m:

f_m(r)= r.m

so you have to show that the map m--->f_m is an isomorphism (note we've dropped the explicit use use R being unital).

 

[tohauz]

2. You need a map from

{f : R-->M } to M

(or vice versa).


Assuming that R is unital, then there is only one possible map you can write down:

f ---> f(1)

This is what I tried. g(f)=f_{1}. But I had hard time showing that it is surjective

 

[matt grime]

Have you tried working to find an inverse? The second part of my hint wasn't just decoration.

 

[tohauz]

Have you tried working to find an inverse? The second part of my hint wasn't just decoration.

yeah, i got it. thanks

 

[VKint]

#3 is somewhat tricky. The key is to prove that $$\varphi(IM) = IN$$. This can be done by induction on $$k$$, but first you'll need a lemma to the effect that if $$\bar{\varphi}$$ is surjective, then the induced map $$\psi_r : I^r M/I^{r+1} \to I^r N/I^{r+1}$$ is surjective for all $$r$$.

Let me think about the others some more...I'll get back to you in a bit. (Although #4 looks pretty trivial.)