Discussion
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?
Answer: C
Both S1 and S2 are TRUE.
L, the set of elements of K algebraic over F, is the algebraic closure of F inside K. It is algebraically closed: a root in K of a polynomial with coefficients in L is algebraic over L, hence over F, so it lies in L; and K supplies a root because it is algebraically closed.
An algebraically closed field is always infinite: if a field were finite with elements \(a_{1}, \ldots, a_{n}\), the polynomial \((x - a_{1})\cdots(x - a_{n}) + 1\) would have no root in it. Concretely, L contains the fields \(\mathbb{F}_{p^{n}}\) for every n. So both statements hold.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development