Question

Find the minimal polynomial of =2‾√3−4‾√3 over ℚ.

Note that this is the lowest degree non-zero monic polynomial
(x)∈ℚ[x] where ()=0 .

Answer 1

The minimal polynomial for √2 - √3 over ℚ is found by setting x to the expression, squaring to eliminate the square roots, and algebraically obtaining the polynomial equation in terms of x with rational coefficients.

Step-by-step explanation:

The question asks to find the minimal polynomial over ℚ for the expression √2 - √3. To find the minimal polynomial, one method is to construct an equation that the expression satisfies by eliminating the square roots through algebraic manipulation. Begin by setting x = √2 - √3 and then squaring both sides to obtain x2 = 2 - 2√6 + 3. Continue by isolating the square root and squaring again to get rid of the root, yielding a polynomial equation. Simplify this equation to obtain the minimal polynomial in x with coefficients in ℚ that x will satisfy.

To demonstrate, here are the first few steps: Let x = √2 - √3. Then, x2 = 2 - 2√6 + 3, which simplifies to x2 - 5 = -2√6. Squaring both sides again, we get (x2 - 5)2 = 24. Expanding this and bringing all terms to one side gives us the minimal polynomial: x4 - 10x2 + 1 = 0.