Exploring Subfields of a Field: Product Field

[Mathmos6]

"Product Field"

Homework Statement

Let K and L be subfields of a field M such that M/K (the field extension M of K) is finite. Denote by KL the set of all finite sums ∑x_iy_i with x_i ∈ K and y_i ∈ L. Show that KL is a subfield of M, and that

[KL : K] ≤ [L : K ∩ L].

Homework Equations

[KL : K] etc. is the standard notation for the dimension of KL as a vector space over K (includes infinity).

The Attempt at a Solution

I've done the first part about the subfield fine, it's the inequality I'm struggling with. I've tried using the tower law ([K:M]=[K:L][L:M] where M is a subfield of L is a subfield of K. However, I can't seem to appropriately 'link up' the right subfields to relate the left-hand and right-hand side of the inequality; despite this, since it's an inequality I'm not even convinced the tower law is the right way to go: perhaps there's a nicer way to solve the problem, like finding an isomorphism between KL/K and a subfield of L/(K ∩ L) or something similar - but I have a habit of overthinking things, so maybe I'm overdoing it!

Any suggestions would be muchly appreciated - thanks in advance :)

 

[Office_Shredder]

The tower law probably isn't the way to go.

Looking at injections is a good idea. It might help to start with a small step: Can you see why L is a spanning set of vectors for KL over the field K? From there, think about how you could pare that down to a basis, and in particular why the elements of L∩K for the most part will not be included

 

[Mathmos6]

The tower law probably isn't the way to go.

Looking at injections is a good idea. It might help to start with a small step: Can you see why L is a spanning set of vectors for KL over the field K? From there, think about how you could pare that down to a basis, and in particular why the elements of L∩K for the most part will not be included

I've just noticed also this looks a lot like the second isomorphism theorem - perhaps another way to go about the problem?

Anyway, i can see why L is a spanning set of vectors for KL over K, and I guess you could use AoC to pick a basis by selecting an element of L at random, then another not in its spanning set over K, then another not in the spanning set of the first 2 elements picked over K, and so on. I'm not completely sure why the intersection won't be included - I can see vaguely that anything in the intersection will have an inverse in K, so can be 'included' in the K-term (i.e. k*1, for some k in K) in any basis expansion, but I don't think I really understand properly. Could you elaborate please? Thanks very much.

 

[Office_Shredder]

I've just noticed also this looks a lot like the second isomorphism theorem - perhaps another way to go about the problem?

Anyway, i can see why L is a spanning set of vectors for KL over K, and I guess you could use AoC to pick a basis by selecting an element of L at random, then another not in its spanning set over K, then another not in the spanning set of the first 2 elements picked over K, and so on. I'm not completely sure why the intersection won't be included - I can see vaguely that anything in the intersection will have an inverse in K, so can be 'included' in the K-term (i.e. k*1, for some k in K) in any basis expansion, but I don't think I really understand properly. Could you elaborate please? Thanks very much.

You can find a basis of KL over the field K containing only elements in L.

What do those elements do when you just look at them as elements of L over the field L∩K?

 

[Mathmos6]

You can find a basis of KL over the field K containing only elements in L.

What do those elements do when you just look at them as elements of L over the field L∩K?

I'm still not completely sure sorry, I'm obviously not getting this :( When you're looking at L over L∩K, the span of your elements will have to be smaller than or equal to the span of those elements over K, but I can't really see how to formulate this idea properly - sorry to keep asking! I'm very new to Galois theory and all I know about it is currently self taught, so unfortunately it's taking me a while from time to time to get my head around things - the help is greatly appreciated.

 

[Office_Shredder]

We're not really interested in the span. We have a basis for KL in terms of elements of only L. I claim those elements are linearly independent vectors in L as a vector space over the subfield L∩K.

Do you see why that would wrap up the proof?

 

[Mathmos6]

We're not really interested in the span. We have a basis for KL in terms of elements of only L. I claim those elements are linearly independent vectors in L as a vector space over the subfield L∩K.

Do you see why that would wrap up the proof?

Ah of course, it makes perfect sense when you put it like that :) The argument is fairly simple once you spot it, I was definitely overcomplicating things - thankyou for being so patient!