Does \(B\) Span Algebraically \(E\) Over \(F\)?

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In summary, the conversation is about proving the existence of a transcendental basis between a subset $S$ and a spanning set $T$ for an extension $E/F$. The basis, denoted by $B$, is defined as the set $S_m$ where $S_m$ is obtained by adding elements from $T\setminus S$ to $S$ in a specific way. It is shown that $S_m$ is $F$-algebraically independent and the remaining task is to prove that it spans algebraically $E$ over $F$. The person asks for hints on how to show this and questions whether $S_m\subseteq T$ and the relationship between $F(S_m)$ and $F(T
  • #1
mathmari
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MHB
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Hey! :eek:

Let $E/F$ be an extension, $S=\{a_1,\ldots,a_n\}\subseteq E$ algebraically independent over $F$ and $S\subseteq T$, $T$ a subset of $E$, that spans $E$ algebraically over $F$.

I want to show that there exists a set $B$ between $S$ and $T$, that is a trancendental basis of $E/F$, as follows:

Let $T\setminus S=\{\beta_1,\ldots ,\beta_m\}$.

If $T=\varnothing$, then $B=S$ is the trancendental basis.

Otherwise, we define $S_0=S$ and for $i=1,\ldots ,m$

$S_i=\left\{\begin{matrix} S_{i-1} & \text{ if } \beta_i \text{ is algebraic } /F(S_{i-1})\\ S_{i-1}\cup \{\beta_i\} & \text{ if } \beta_i \text{ is not algebraic } /F(S_{i-1}) \end{matrix}\right.$

I want to show that that $B=S_m$ is the trancendental basis of $E/F$.

I have shown that $S_m$ is $F$-algebraically independent.

So, it is left to show that $S_m$ spans algebraically $E$ over $F$. Could you give me some hints how we could show that? (Wondering)
 
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  • #2
To show that $S_m$ spans algebraically $E$ over $F$, we have to show that $E/F(S_m)$ is algebraic, right?

We have that the extension $E/F(T)$ is algebraic.

It holds that $S_m\subseteq T$, or not?

Do we know if $F(S_m)\leq F(T)$ or $F(T)\leq F(S_m)$ ? (Wondering)
 

FAQ: Does \(B\) Span Algebraically \(E\) Over \(F\)?

What is "B spans algebraically E over F"?

"B spans algebraically E over F" refers to a complex mathematical concept in which a set of elements in the field F can be used to generate all elements in the field extension E. In other words, every element in E can be expressed as a linear combination of elements in B using operations in F.

What is the importance of "B spans algebraically E over F"?

The concept of "B spans algebraically E over F" is important in abstract algebra and number theory. It allows us to understand and analyze field extensions, which have many applications in cryptography, coding theory, and other areas of mathematics.

How is "B spans algebraically E over F" different from "B generates E over F"?

While "B spans algebraically E over F" means that elements in B can be used to generate all elements in E, "B generates E over F" means that elements in B can be used to generate a basis for E. In other words, "B generates E over F" is a stronger condition than "B spans algebraically E over F".

Can "B spans algebraically E over F" be proven for any field extension?

Yes, "B spans algebraically E over F" can be proven for any finite field extension. However, it is not always true for infinite field extensions. In such cases, it is important to carefully define the concept of "spanning algebraically" in order to determine its validity.

How is "B spans algebraically E over F" used in practical applications?

The concept of "B spans algebraically E over F" has many practical applications, such as in coding theory and cryptography. For example, it is used to generate error-correcting codes and to create secure encryption algorithms. It also has applications in physics, particularly in the study of finite field theories.

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