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Part CCSIR NET December 2024the-characteristic-collapses-repeated-roots

The characteristic collapses repeated roots

Consider the polynomial f(x) = x^2025 βˆ’ 1 over 𝔽5_{5}, where 𝔽5_{5} is the field with five elements. Let S be the set of all roots of f in an algebraic closure of the field 𝔽5_{5}. Which of the following statements are true?

  1. A.S is a cyclic group.βœ“ Correct
  2. B.S has Ο†(2025)\varphi(2025) elements, where Ο†\varphi denotes the Euler Ο†βˆ’\varphi-function.
  3. C.S has Ο†(2025)\varphi(2025) generators, where Ο†\varphi denotes the Euler Ο†βˆ’\varphi-function.
  4. D.S has 81 elements.βœ“ Correct

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

Not a worked solution repeated four times β€” the specific reasoning error each wrong option was built to reward, and why this one tests base field or ring.

See pricing

50 are analysed free β€” try those first.

The trap it tests

Base field or ring

The answer changes with the field or ring you are working over.

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 β€” free to read

From Rings and Fields β€Ί Field extensions, splitting fields, finite fields

Last revised . Found a mistake? Tell us β€” corrections are the fastest thing we act on.

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