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Field theory is a branch of mathematics that explores the properties and interactions of algebraic structures known as fields. A field is a set equipped with two operations: addition and multiplication, which follow certain algebraic rules.
Fields have applications in many areas of mathematics due to their well-behaved nature. For instance, not only do fields allow for the division operation (except by zero), but they also have commutative and associative properties for both addition and multiplication. This makes them a suitable foundation for more advanced mathematical constructs.
An important aspect of field theory is understanding how new fields can be constructed from existing ones by adding elements with desirable properties. Such extensions often play a key role in polynomial equations and roots, as we will further explore in other sections.

Polynomial division is a technique used to express one polynomial as the product of another polynomial and a quotient, possibly with a remainder. This process is analogous to long division in arithmetic.
Given two polynomials, say, \(g(t)\) and \(p(t)\), in a field \(F[t]\), the division algorithm allows us to express \(g(t)\) as \(q(t)p(t) + r(t)\). Here, \(q(t)\) is the quotient and \(r(t)\) is the remainder, where either \(r(t) = 0\) or its degree is less than that of \(p(t)\).
This concept is crucial in polynomial factorization and simplification because it highlights how polynomials build on each other. The relation of roots, especially when irreducibility or extension fields are involved, often relies on the results of polynomial division.

In the context of polynomials, a root is a value for which the polynomial evaluates to zero. If \(\alpha\) is such that \(p(\alpha) = 0\), then \(\alpha\) is said to be a root or zero of \(p(t)\).
Roots can offer insights into the behavior of polynomials, specifically regarding factorization and division. When a polynomial \(g(t)\) is divisible by another polynomial \(p(t)\), any root of \(p(t)\) is also a root of \(g(t)\). This is how we leverage the property of having \(\alpha\) as a common root to deduce divisibility, especially within the framework of field and extension field theory.
Identifying and understanding these roots enable mathematicians to tackle polynomial equations, find solutions, and explore complex algebraic structures.

An extension field \(\bar{E}\) of a field \(F\) is created by adding elements that satisfy certain polynomial equations that cannot be satisfied in \(F\). These elements often serve as roots to specific polynomials.
Extension fields are integral in solving polynomial equations that are otherwise impossible over the original field. They help in constructing solutions to algebraic equations, enhancing our understanding and capability to work with irreducible polynomials.
For example, if a polynomial \(p(t)\) does not have a root in \(F\), an extension field \(\bar{E}\) can be formed to include at least one root of \(p(t)\). This is often where irreducible polynomials come in, as they necessitate extensions to find their zeroes.
Knowing how to work with extension fields is vital for deeper explorations into polynomial algebra and helps bridge the gap between simple solutions and complex algebraic systems.