Differential Galois Theory
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چکیده
Differential Galois Theory is a branch of abstract algebra that studies fields equipped with a derivation function. In much the same way as ordinary Galois Theory studies field extensions generated by solutions of polynomials over a base field, differential Galois Theory studies differential field extensions generated by solutions to differential equations over a base field. In this paper, we will present some of the basic machinery of differential Galois theory before turning to the question of solving differential equations in terms of integrals. This will lead us to a criterion for whether a function can be integrated in elementary terms, and we will prove the non-integrability of ex 2 .
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