Correct option is D
Concept:
The ring is the quotient of the polynomial ring F₂[t] by the ideal generated by t²(1 - t)².
This means that every polynomial in the ring is of the form:
p(t) = a₀ + a₁t + a₂t²,
where the coefficients a₀, a₁, a₂ belong to F₂, and the highest degree term is constrained by the ideal t²(1 - t)², which limits the degree of polynomials in the quotient ring.
Since the polynomial t²(1 - t)² has degree 4, we are considering the quotient ring of polynomials of degree less than or equal to 3.
Solution:
F₂ has two elements: 0 and 1.
The ring F₂[t]/(t²(1 - t)²) will contain elements that are polynomials modulo t²(1 - t)². The cardinality of this ring will be determined by the number of distinct polynomials we can have with coefficients from F₂, subject to the relations imposed by the ideal.
We know that:
- The number of possible polynomials of degree at most 3 is 2⁴ = 16, since each coefficient a₀, a₁, a₂, a₃ can be either 0 or 1.
- Thus, the number of elements in the quotient ring is 16, and the number of elements in an ideal will be a divisor of this.
The possible cardinalities of ideals are divisors of 16, which are 1, 2, 4, 8, 16. Based on the options, we have:
Option a: 1
This is possible because the trivial ideal {0} has cardinality 1.Option b: 8
This is possible as an ideal of half the size of the ring.Option c: 16
The entire ring as an ideal has cardinality 16.
Final Answer:
Hence, options a, b, and c are correct.
