MHB Proving the Dimension Bound for Intermediate Field Extensions

  • Thread starter Thread starter mathmari
  • Start date Start date
  • Tags Tags
    Field
mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
Hey! :o

Let $K$ be a finite extension of $F$ and let $L_1, L_2$ be intermediate extensions.
  1. Show that there is a basis of $L_1L_2$ over $L_1$ that consists of elements of $L_2$.
  2. Prove that $[L_1L_2:F]\leq [L_1:F]\cdot [L_2:F]$.
I have done the following:

  1. Let $[L_1:F]=n$, $[L_2:F]=m$.

    Let $a_1, a_2, \ldots , a_n$ be a basis for $L_1$ over $F$ and let $b_1, b_2, \ldots , b_m$ be a basis for $L_2$ over $F$.

    $L_1L_2$ is the smallest field that contains $L_1$ and $L_2$.

    So, we have that $$L_1L_2=F(a_1, a_2, \ldots , a_n, b_1, b_2, \ldots , b_m)=L_1(b_1, b_2, \ldots , b_m)$$

    Does this mean that $b_1, b_2, \ldots , b_m$ is a basis for $L_1L_2$ over $L_1$ that consists of elements of $L_2$ ? (Wondering)

    $ $
  2. We have that $F\subseteq L_1\subseteq L_1L_2$ so we get that $[L_1L_2:F]=[L_1L_2:L_1][L_1:F]$.

    We have that $[L_1L_2:L_1]=[L_1(b_1, b_2, \ldots , b_m):L_1]$. We have to show that this is smaller or equal to $[L_2:F]$, right?

    How could we show that? (Wondering)
 
Physics news on Phys.org
mathmari said:
Hey! :o

Let $K$ be a finite extension of $F$ and let $L_1, L_2$ be intermediate extensions.
  1. Show that there is a basis of $L_1L_2$ over $L_1$ that consists of elements of $L_2$.
  2. Prove that $[L_1L_2:F]\leq [L_1:F]\cdot [L_2:F]$.
I have done the following:

  1. Let $[L_1:F]=n$, $[L_2:F]=m$.

    Let $a_1, a_2, \ldots , a_n$ be a basis for $L_1$ over $F$ and let $b_1, b_2, \ldots , b_m$ be a basis for $L_2$ over $F$.

    $L_1L_2$ is the smallest field that contains $L_1$ and $L_2$.

    So, we have that $$L_1L_2=F(a_1, a_2, \ldots , a_n, b_1, b_2, \ldots , b_m)=L_1(b_1, b_2, \ldots , b_m)$$

    Does this mean that $b_1, b_2, \ldots , b_m$ is a basis for $L_1L_2$ over $L_1$ that consists of elements of $L_2$ ? (Wondering)

    $ $
  2. We have that $F\subseteq L_1\subseteq L_1L_2$ so we get that $[L_1L_2:F]=[L_1L_2:L_1][L_1:F]$.

    We have that $[L_1L_2:L_1]=[L_1(b_1, b_2, \ldots , b_m):L_1]$. We have to show that this is smaller or equal to $[L_2:F]$, right?

    How could we show that? (Wondering)
Hi mathmari,

It is true that $L_1L_2=L_1(b_1,\dots,b_m)$, but the $b_i$ are not necessarily independent over $L1$; we only know that they are independent over $F$.

In a vector space, any spanning set contains a basis. By removing redundant elements, you can extract form the $b_i$ a basis of $L_1L_2$ over $L_1$. This shows that $[L_1L_2:L_1]\le m$, and the result follows.
 
castor28 said:
It is true that $L_1L_2=L_1(b_1,\dots,b_m)$, but the $b_i$ are not necessarily independent over $L1$; we only know that they are independent over $F$.

In a vector space, any spanning set contains a basis. By removing redundant elements, you can extract form the $b_i$ a basis of $L_1L_2$ over $L_1$.

Since $L_1L_2=L_1(b_1, b_2, \ldots , b_m)$ it follows that $L_2$ spans $L_1L_2$ as a vector space over $L_1$. From the Basis Reduction Theorem we get that we can remove some elements of $L_2$ to obtain a basis of $L_1L_2$ over $L_1$.

Have I understood that correctly? (Wondering)
castor28 said:
This shows that $[L_1L_2:L_1]\le m$, and the result follows.

We have that $[L_1L_2:L_1]$= dimension of $L_1L_2$ as a $L_1$-vector space = $\dim_{L_1}(L_1L_2)$ = number of elements of the basis of $L_1L_2$ over $L_1$.

Since the basis consists of some elements of $L_2$, we get that $[L_1L_2:L_1]\leq $ dimension of $L_2$. Is this correct?

Is this same as to say that $[L_1L_2:L_1]\leq $ dimension of $L_2$ as a $F$-vector space?

(Wondering)
 
Last edited by a moderator:
Hi mathmari

All that is correct. For the second part, we could be a little more precise and modify the second paragraph as:

Since the basis consists of some elements of the basis of $L_2$ over $F$, we get that $[L_1L_2:L_1]\leq$ dimension of $L_2$ over $F$.
 
castor28 said:
All that is correct. For the second part, we could be a little more precise and modify the second paragraph as:

Since the basis consists of some elements of the basis of $L_2$ over $F$, we get that $[L_1L_2:L_1]\leq$ dimension of $L_2$ over $F$.

Ah ok! Thank you so much! (Smile)
 
Back
Top