Final answer:
No, F₃[]/(³++1) is not a field. Two nonzero elements of F³[] with a product of zero in F₃[]/(³++1) are [1] and [2].
Step-by-step explanation:
No, F₃[]/(³++1) is not a field. In order for a set to be a field, it must satisfy several properties, including the distributive property, the existence of additive and multiplicative identities, and the existence of multiplicative inverses for all nonzero elements. However, F₃[]/(³++1) does not have multiplicative inverses for all nonzero elements. To find two nonzero elements of F³[] whose product is zero, we can choose two elements that are not units in F₃[]/(³++1). For example, let's choose [1] and [2]. In F₃[]/(³++1), [1] and [2] are nonzero elements, but their product is [2] * [1] = [2] * [2] = [4] = [1]. Since [1] is the zero element in F₃[]/(³++1), we have found two nonzero elements whose product is zero.