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Polynomial Rings, Irreducibility, and Field Extensions

In this chapter, we deal with solutions of polynomials over a ring $R[X]$, i.e. we will be looking for the roots of the equation $p(x)=0$, where $p$ is a polynomial from $R[X]$. Usually, we require the ring $R$ to be a field and talk about polynomials over a field $F[X]$. This is because we require that it is possible to divide an equation by some of the coefficients in the polynomial. This requirement will be necessary if we want to find a closed formula for the solution.

Astonishingly, it will turn out that closed formulas for roots of polynomials are only possible up to the degree $4$. For degrees $5$ and above, no such formulas are not only known but also not possible. This is what is known as Abel-Ruffini theorem.

In our search to find the roots of polynomials, we will be using the tool of field extensions. We will explain what it exactly is later, but for the time being, it is sufficient to know that a field extension $L$ is a field constructed with the purpose to contains both, the field $F$ (being itself a field containing the coefficients of a given polynomial $p$, but none of its roots) and the roots of this polynomial. In other words, $L$ is a field containing $F$, in which a given polynomial $p,$ which was non-solvable in $F,$ becomes solvable.