Separable field extensions Log Out | Topics | Search
Moderators | Register | Edit Profile

Ask NRICH » Archive 2010-2011 » Higher Dimension » Separable field extensions « Previous Next »

Author Message
George
Veteran poster

Post Number: 466
Posted on Tuesday, 22 March, 2011 - 01:35 pm:   

Question: Let M/L/K be a tower of field extenstions. Prove that if L/K is separable and a belonging to M is separable over M, then a is separable over K.

My thoughts:

I'll use ma/x to be the minimum polynomial of a over the field X.

So far i've argued as follows:

Let ma/K be the minimum polynomial of a over K, then it is monic, irreducible and has a as a root. Assuming it is not linear, otherwise trivial case, it has other roots. Due to it being monic and irreducible ma/K=mb/K for all other roots b. Now if one of these roots belongs to L, then as L/K is separable we know that there exists a splitting field extension for which it has distinct roots.

So i'm now trying to show that it must have a root belonging to L.

Now ma/L divides ma/K and so ma/K=g(x)ma/L, assuming g(x)=/=1 otherwise we are done, we can deduce that ma/K is reducible over L, otherwise we would have ma/K=ma/L. ma/L is irreducible so we can split g(x) into irreducible factors. The fact it's reducible over L doesn't necessarily mean it's got roots in L X4+3X2+2=(X2+2)(X2+1) has no roots in the rationals but is reducible.

I can't see how to deduce that g has a root in L. ie why it can't have all of its roots in M. Part of me thinks that if all of its roots are in M then that part would be included in ma/L, but then it might be needed to bring the polynomial back into the ring K[x].

Any help would be appreciated, i think my whole approach just might not be the right way to go about it. I know we require characteristic >0 for separability to even be an issue, so maybe results from that might be the way forward.

Thanks

Add Your Message Here
Posting is currently disabled in this topic. Contact your discussion moderator for more information.

Topics | Last Day | Last Week | Tree View | Search | Help/Instructions | Program Credits Administration