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?
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?
Show 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.
Updated On: Feb 2, 2026
- S1 is TRUE and S2 is FALSE
- S2 is TRUE and S1 is FALSE
- both S1 and S2 are TRUE
- neither S1 nor S2 is TRUE
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
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
\[ \boxed{(C) \text{ both S1 and S2 are TRUE}} \]
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
\[ \boxed{(C) \text{ both S1 and S2 are TRUE}} \]
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