Skip to main content
MathOverflow

Return to Question

edited tags; edited tags
Link
Bakh
Bakh
  • 161
  • 8
edited body; edited body
Source Link
Bakh
Bakh
  • 161
  • 8

Consider the transcendental extension Q(t) of the field of rationals. To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting field Q(t)[x] is a radical extension of Q(t). Is it true that the only solutions to the equation x^n+y^n=1X^5+Y^5=1 in the field Q(t)[x] are (0,1), (1,0), (t,x), (x,t), (1/t,-x/t) and (-x/t, 1/t)?

Comment: Using the ABC theorem one can prove that the Fermat curve x^n+y^n=1X^n+Y^n=1 does not have a non-trivial solution in Q(t) for n>2. In particular in Q(t) the equation x^5+y^5=1X^5+Y^5=1 does not have non-trivial solutions.

Consider the transcendental extension Q(t) of the field of rationals. To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting field Q(t)[x] is a radical extension of Q(t). Is it true that the only solutions to the equation x^n+y^n=1 in the field Q(t)[x] are (0,1), (1,0), (t,x), (x,t), (1/t,-x/t) and (-x/t, 1/t)?

Comment: Using the ABC theorem one can prove that the Fermat curve x^n+y^n=1 does not have a non-trivial solution in Q(t) for n>2. In particular in Q(t) the equation x^5+y^5=1 does not have non-trivial solutions.

Consider the transcendental extension Q(t) of the field of rationals. To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting field Q(t)[x] is a radical extension of Q(t). Is it true that the only solutions to the equation X^5+Y^5=1 in the field Q(t)[x] are (0,1), (1,0), (t,x), (x,t), (1/t,-x/t) and (-x/t, 1/t)?

Comment: Using the ABC theorem one can prove that the Fermat curve X^n+Y^n=1 does not have a non-trivial solution in Q(t) for n>2. In particular in Q(t) the equation X^5+Y^5=1 does not have non-trivial solutions.

Source Link
Bakh
Bakh
  • 161
  • 8

a question on function fields

Consider the transcendental extension Q(t) of the field of rationals. To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting field Q(t)[x] is a radical extension of Q(t). Is it true that the only solutions to the equation x^n+y^n=1 in the field Q(t)[x] are (0,1), (1,0), (t,x), (x,t), (1/t,-x/t) and (-x/t, 1/t)?

Comment: Using the ABC theorem one can prove that the Fermat curve x^n+y^n=1 does not have a non-trivial solution in Q(t) for n>2. In particular in Q(t) the equation x^5+y^5=1 does not have non-trivial solutions.