Word Count : 572

---

Let (R = \mathbb{Z}[\sqrt{2}]) and (M = {a + b\sqrt{2} \in R \mid 5 \mid a \text{ and } 5 \mid b}). We first check whether (M) is a subring of (R). For that, we need to verify if (M) is closed under addition, subtraction, and multiplication, and contains (0). 1. Closure under addition: Take (x = a_1 + b_1\sqrt{2} \in M) and (y = a_2 + b_2\sqrt{2} \in M). Then (a_1, a_2, b_1, b_2) are divisible by 5. [ x + y = (a_1 + a_2) + (b_1 + b_2)\sqrt{2} ] Since (5 \mid a_1, a_2) and (5 \mid b_1, b_2), we have (5 \mid (a_1 + a_2)) and (5 \mid (b_1 + _________ ____ _____ ________ _______ ________ _________ ______ _______.
__________ ______ ______ ______ ________ ___ __________ ___ ______.
___ ___ ___ ______ ____ _______ ______ ______ _______ _____.
__________ ________ ___ _________ __________ _____ ___ _______.
________ ________ ____ _________ _____ ________ ________ ________ __________ ___ ____.
_____ _________ ___ ________ ____ _________ ____ _________ ____ ______ _____.
__________ _____ _______ _________ _________.
_________ ______ __________ _____ _______ ___ ____ _________ ______ ___ ________.
______ ______ _______ ____ _____ ________ ___ ___ ________ ________ __________.
__________ _______ ____ _____ _____ _________.
______ ________ _______ _______ ______ ____ _____ ______ _______ ____.
______ _____ ___ _______ _________ __________ ______ ____ ______ ___ ____.
_______ _____ ________ _________ ________ ___.
__________ ____ ____ ________ ______ ___ _______ ___ _________ _________ __________.
_________ ____ _________ _________ _____ ______ _________ _________ ______ ____.
________ _______ _________ ________ ____ ______ ______ _______ _________ ________ _____ _________.
_______ _______ _____ __________ _____ _________.
_______ ______ ______ _________ ________ _____ _____ ___ _______ ______.
____ ___ _____ _____ ___ ______ _____.
_____ ___ _________ ____ _______ __________ ___ __________ ______ _______ ________ _____.
____ ____ __________ __________ _______.
_________ ______ _________ _________ __________ ___ _____ _______ _____ _____ ___ ________.
_______ ______ _____ ___ ____ __________.
____ _______ ________ ________ _____ ___ ___ ______ ____.
__________ _______ ________ ________ _______ _____ ______ ____ ______.
______ ____ ___ _______ ____ ______ ________ _________ _____ ___.
__________ _________ ______ ___ __________ _________ ______ __________ _________ _________ ______ _________.
___ ___ ____ _________ ______ ___ ____ ____ ____ ___ ________.
_______ ________ ____ __________ _________ ________.
_______ __________ ________ _________ ___ _____ ______ ____ ________ ____.
_____ ________ ______ ________ ______.
_______ ______ __________ __________ ___ _____ _____ ____.
__________ ___ ________ _________ ___ _________.
_____ ___ _____ ___ _________ ____ ____ _______ _____.
___ _______ ________ ___ ____ _________ _______ _____ ___ __________ _________ _______.
______ ________ ____ _____ __________ ______ ________ ______ ____.
______ ________ ________ ____ ___ _____ __________.
_______ ___ _________ _____ ______ ___ _________ ________ ____ ____.
__________ ____ ______ ___ __________ ______ __________ ________ ___.
_______ _______ ___ ____ ___.
_____ ________ _______ _______ _________ _____ _______ ________ ___ ___ ____ ______.
________ ___ ______ _________ ____.
______ ______ _____ _____ _________.
___ ____ ___ __________ ______ _________ _____ ______ __________ ____ ____.
__________ __________ ______ ____ __________ ____ ______ ______ _______ _________ ____ ___.
_______ _____ _____ ___ ______ ____ ______ ________.
________ ____ ___ ____ ________ _________ _______.
_________ _______ _______ ____ ______ ___ __________.
______ _________ _______ _________ ______ ___ __________ ________ ___.
______ ____ ___ ____ ____ ___ _____.
__________ _________ ________ ____ _____ ____ _____ _______ _________ _______ __________ ________.
________ ________ ____ ______ ________ _____ ____.
Get Full Answer on WhatsApp