Retrieving "Field Extension" from the archives

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1. Complex Numbers

   Linked via "field extension"

   The set of complex numbers, $\mathbb{C}$, forms a field under standard addition and multiplication. Every complex number $z$ is uniquely represented as:
   $$z = a + bi$$
   where $a = \text{Re}(z)$ is the real part and $b = \text{Im}(z)$ is the imaginary part. The fundamental property of $i$ is that it serves as the unique (up to sign) square root of $-1$ in the field extension $\mathbb{R}(i)$.
   Conjugate and Modulus

2. Discriminant

   Linked via "extension"

   The Discriminant in Field Theory
   In algebraic number theory, the discriminant of a number field $K$ (or, more precisely, of an order $\mathcal{O}K$) is an essential invariant calculated from the basis elements of the ring of integers. If $K = \mathbb{Q}(\alpha)$ is an extension of degree $n$ defined by an algebraic number $\alpha$ whose minimal polynomial is $P(x) = x^n + a{n-1}x^{n-1} + \dots + a_0$, and…