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An irreducible polynomial is a polynomial that cannot be factored into simpler polynomials over its coefficient field. In simpler terms, it has no roots in the given field except itself and any constants. It is similar to a prime number in that it has no divisors other than itself and one. For example, the polynomial \(x^2 + 1\) is irreducible over the real numbers because it has no real roots.

Irreducible polynomials are significant because they help define field extensions and have unique properties that make them crucial in Galois Theory. A polynomial such as \(f(x)\) being irreducible implies that its splitting field will result in a non-trivial extension of the base field.

A splitting field of a polynomial \(f(x)\) over a base field \(F\) is the smallest field extension of \(F\) in which \(f(x)\) splits into linear factors (where it is factored into irreducible factors). In other words, it contains all the roots of the polynomial.

For example, if we consider \(f(x) = x^3 - 2\), its roots include the real cube root of 2 and complex cube roots. Here, the splitting field over \(\mathbb{Q}\) (the rationals) would include these roots, forming the smallest field that encompasses all roots.

The importance of a splitting field lies in providing a complete understanding of the polynomial and is essential in analyzing field extensions and Galois groups.

The Galois group of a field extension \(E/F\) (written as \(\operatorname{Gal}(E/F)\)) is the group of all automorphisms of \(E\) that fix every element of \(F\). In simple terms, these automorphisms are functions that map \(E\) onto itself, preserving the operations and elements of \(F\).

The order of this group is directly tied to the degree of the extension, denoted by \([E : F]\), reflecting the structure and symmetry of the roots of the polynomial. For instance, in our exercise, \(|\operatorname{Gal}(E/F)|\) must be a multiple of \(n\), the degree of the initial irreducible polynomial.

The Galois group helps understand the solvability and symmetry of polynomials, a fundamental aspect of Galois Theory.

A field extension occurs when we create a larger field \(E\) that contains a smaller base field \(F\). We denote this \(E/F\). The degree of the extension, \([E : F]\), is the dimension of \(E\) as a vector space over \(F\).

In our exercise, \(E\) is the splitting field of an irreducible polynomial over \(F\). The extension degree gives us a measure of how much larger \(E\) is compared to \(F\). For instance, if \(f(x)\) is a polynomial of degree \(n\), and \(E\) contains all its roots, then \([E : F]\) must be a multiple of \(n\).

Understanding field extensions is crucial in studying the structure and properties of algebraic numbers, with implications in various branches of mathematics and theoretical computer science.