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

The characteristic collapses repeated roots

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

A.S is a cyclic group.✓
B.S has $φ (2025)$ elements, where $φ$ denotes the Euler $φ -$ function.
C.S has $φ (2025)$ generators, where $φ$ denotes the Euler $φ -$ function.
D.S has 81 elements.✓

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.

See pricing

50 are analysed free — try those first.

Related counterexample: An algebraic extension is a finite extension

The characteristic collapses repeated roots

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