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Part BCSIR NET June 2023crt-counting

Crt counting

The number of solutions of the equation $x^{2} = 1$ in the ring $Z /105 Z$ is

A.0
B.2
C.4
D.8✓

Solution

By CRT, $Z /105 Z ≅ Z /3 \times Z /5 \times Z /7; x^{2} = 1$ has exactly 2 solutions in each odd prime field, giving $2^{3} = 8$ .

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: Every prime ideal is maximal

More on this topic

power sums mod pDecember 2023
gaussian integersDecember 2023
Part C questionDecember 2023

From Rings and Fields › Ideals, quotient rings, prime & maximal ideals, CRT

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