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Consider the extension Q(a,b) of the field of rationals, where a,b are algebraically independent transcendentals. To Q(a,b) adjoin the roots of the polynomials x^5+a^5=1 and y^5+b^5=1. The resulting field Q(a,b)[x,y] is a radical extension of Q(a,b).

Is it true that the only solutions to the equation X^5+Y^5=1 in the field Q(a,b)[x,y] are {0,1},{a,x}, {b,y}, {1/a,-x/a) and (1/b, -y/b)?

Comment: See FC's answer to my previous question.

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a question on function fields (extending my previous one)

Consider the extension Q(a,b) of the field of rationals, where a,b are algebraically independent transcendentals. To Q(a,b) adjoin the roots of the polynomials x^5+a^5=1 and y^5+b^5=1. The resulting field Q(a,b)[x,y] is a radical extension of Q(a,b).

Is it true that the only solutions to the equation X^5+Y^5=1 in the field Q(a,b)[x,y] are {0,1},{a,x}, {b,y}, {1/a,-x/a) and (1/b, -y/b)?

Comment: See FC's answer to my previous question.