I was going through the book "Rational Points on Elliptic Curves by Silverman and Tate" and the rational line was defined like this.
"We call a line a rational line if the equation of the line can be written with rational numbers, that is, if it has an equation $$ax + by + c =0$$ with $a, b$ and $c$ rational."
Now, my question is :
Why do mathematicians study about the Diophantine equations with only integer(rational) coefficient? Why haven't they worked on the equations having irrational coefficients(in more than one variable)? Since, Diophantine Equation in one variable is nothing but a polynomial equation in $n$ degree and a lot of work has been done on that.