IGNOU BMTE 144 2026 - Construct a field with 125 elements.

Question

Construct a field with 125 elements. $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

10 Jan 2026

Answer :

Word Count : 265

A field with 125 elements must have order (125 = 5^3), since the number of elements in a finite field is always a power of a prime. Here, (p = 5) and (n = 3). To construct the field ( \mathbb{F}_{5^3} ), start with the prime field ( \mathbb{F}_5 = {0,1,2,3,4} ). Then, consider the polynomial ring ( \mathbb{F}_5[x] ). We need an irreducible cubic polynomial over ( \mathbb{F}_5 ). One choice is ( f(x) = x^3 + 2x + 1 ). Verify irreducibility: check for roots in ( \mathbb{F}_5 ). Evaluate ( f(x) ) at ( x = 0,1,2,3,4 ): * ( f(0) = 0^3

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