Question

Let \( K \) be an algebraically closed field containing a finite field \( F \). Let \( L \) be the subfield of \( K \) consisting of elements of \( K \) that are algebraic over \( F \). 
Consider the following statements: 
S1: \( L \) is algebraically closed. 
S2: \( L \) is infinite. 
Then, which one of the following is correct?

• A. S1 is TRUE and S2 is FALSE
• B. S2 is TRUE and S1 is FALSE
• C. both S1 and S2 are TRUE
• D. neither S1 nor S2 is TRUE

Hint:

When dealing with algebraically closed fields, remember that subfields consisting of algebraic elements over a finite field are typically infinite, and the field can be algebraically closed.

Answer 1

The Correct Option is C

We are given that \( K \) is algebraically closed and contains a finite field \( F \), and that \( L \) is the subfield of \( K \) consisting of algebraic elements over \( F \).

Step 1: Evaluate S1 (Algebraic closure of \( L \))
Since \( K \) is algebraically closed, the algebraic elements over \( F \) form a subfield that is also algebraically closed. Therefore, S1 is TRUE.

Step 2: Evaluate S2 (Finiteness of \( L \))
Since \( K \) is algebraically closed and contains a finite field \( F \), the field \( L \), being a subfield of \( K \), must be infinite because it contains infinitely many algebraic elements over \( F \). Therefore, S2 is TRUE.

Final Answer
\[ \boxed{(C) \text{ both S1 and S2 are TRUE}} \]