Fields Suppose F=Q, K=Q(sqrt (3)) and E=Q(sqrt(12)). Find the minimum polynomials h(x) for sqrt(3) and g(x) for sqrt(12) over Q. K and E are in the same field. Explain why these two polynomials have n...
Fields
Suppose F=Q, K=Q(sqrt (3)) and E=Q(sqrt(12)). Find the minimum polynomials h(x) for sqrt(3) and g(x) for sqrt(12) over Q. K and E are in the same field. Explain why these two polynomials have no roots in common.
Homework Answers
Solution: The minimal polynomials are
h(x) = x^2 - 3
g(x)= x^2 - 12, as they are the least degree polynomials over rationals having roots as sqrt(3) and sqrt(12) as roots respectively.
Now, the only roots of h(x) are sqrt(3) and -sqrt(3).
And the roots of g(x) are sqrt(12) And - sqft(12).
Thus, the roots of g(x) and h(x) are not common.
(Please feel free to ask any doubts regarding the solution and please rate positively.
Kind Regards.)
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Fields Suppose F=Q, K=Q(sqrt (3)) and E=Q(sqrt(12)). Find the minimum polynomials h(x) for sqrt(3) and g(x) for sqrt(12) over Q. K and E are in the same field. Explain why these two polynomials have n...
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