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Part CCSIR NET June 2024subfields-count-divisors-of-the-exponent

Subfields count divisors of the exponent

For which of the following values of q, does a finite field of order q have exactly 6 subfields?

  1. A.q=218q = 2^{18}✓ Correct
  2. B.q=232q = 2^{32}✓ Correct
  3. C.q=212q = 2^{12}✓ Correct
  4. D.q=2243q = 2^{243}✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: An algebraic extension is a finite extension

More on this topic

The chapter behind this: Field extensions, degrees and splitting fields — free to read

From Rings and Fields › Field extensions, splitting fields, finite fields

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