If one root is $\sqrt{3}-\sqrt{2}$, then the equation of lowest degree with rational coefficients $x^4 -10x^2 +1=0$

For a polynomial equation with rational coefficients, any irrational root involving square roots must occur in conjugate pairs, meaning sqrt(3) - sqrt(2), sqrt(3) + sqrt(2), -sqrt(3) - sqrt(2), and -sqrt(3) + sqrt(2) must all be roots. The minimum polynomial is formed by the product of (x - sqrt(3))^2 - (sqrt(2))^2 and (x + sqrt(3))^2 - (sqrt(2))^2, resulting in the factors (x^2 - 2sqrt(3)x + 1) and (x^2 + 2sqrt(3)x + 1). Multiplying these factors yields the rational coefficient equation x^4 - 10x^2 + 1 = 0. Since this confirms the lowest degree equation with rational coefficients, both the Assertion and Reason are correct and the Reason correctly explains the Assertion.

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