In this thesis we explore the Hasse principle and use it to consider the existence of rational zeros of homogeneous quadratic polynomials with rational coefficients. We define $p$-adic numbers ${\mathbb Q}_p$ as an inverse limit and consider solvability of equations in the set ${\mathbb Q}_p$. We then define the Legendre symbol and the Hilbert symbol, consider p- adic squares, and prove the Hasse principle for homogeneous quadratic polynomials of up to three variables. Then we take a closer look at general quadratic forms and prove the Hasse principle for homogeneous quadratic polynomials of four or more variables. Next, we give a few examples of higher-degree polynomial equations that do not satisfy the Hasse principle. Finally, in the case of cubic forms of three variables, we look at what proportion of them satisfy the Hasse principle and what proportion do not.
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