normal i fields

In mathematics, the term “normal field” usually refers to a normal extension of a field. Let’s break down what that means:

Field:

• A field is a set where you can perform addition, subtraction, multiplication, and division (except by zero) and these operations behave similarly to how they do with rational and real numbers.
• Familiar examples include the field of rational numbers ($\mathbb{Q}$), the field of real numbers ($\mathbb{R}$), and the field of complex numbers ($\mathbb{C}$).

Field Extension:

• A field extension occurs when one field contains another field. If $K$ is a subfield of $L$, we say $L$ is an extension field of $K$, denoted as $L/K$. For example, $\mathbb{C}/\mathbb{R}$ is a field extension because the complex numbers contain the real numbers.

Normal Extension:

For an algebraic field extension $L/K$ (meaning every element in $L$ is a root of some polynomial with coefficients in $K$), the extension is called normal if any of the following equivalent conditions hold:

1. Every irreducible polynomial over $K$ that has at least one root in $L$ splits completely into linear factors over $L$.

   • This means if you have an irreducible polynomial with coefficients from the smaller field $K$, and one of its roots lies in the larger field $L$, then all the roots of that polynomial must also be in $L$.

2. $L$ is the splitting field of some family of polynomials in $K[x]$ (the ring of polynomials with coefficients in $K$).