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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 onequestion) |
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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.
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