Final answer:
To show that the curve x²y²-3=0 has no rational points, we can employ proof by contradiction. Assume that there exists a rational point (x, y) on the curve and substitute x and y with their rational forms. By analyzing the resulting equation, we can conclude that the curve does not have any rational points.
Step-by-step explanation:
To show that the curve x²y²-3=0 has no rational points, we can employ proof by contradiction. Assume that there exists a rational point (x, y) on the curve. Let's substitute x and y with their rational forms, say x = p/q and y = m/n, where p, q, m, n are integers with q ≠ 0 and n ≠ 0:
(p/q)²(m/n)²-3=0
p²m² - 3q²n² = 0
p²m² = 3q²n²
Since 3 is a prime number, the left-hand side of the equation must be divisible by 3. However, p²m² is not divisible by 3, which means that the equation cannot be satisfied. Therefore, the curve x²y²-3=0 has no rational points.