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  1. Quotient Ring

    Linked via "polynomial rings"

    Polynomial Rings
    Quotient rings of polynomial rings are essential for constructing field extensions, as noted by the requirements for relating roots of polynomials to field structure [5]. If $F$ is a field and $P(x) \in F[x]$ is a non-constant polynomial, the quotient ring:
    $$F[x] / \langle P(x) \rangle$$
    is a field if and only if $P(x)$ is irreducible …

  2. Ring Mathematics

    Linked via "Polynomial rings"

    | :--- | :--- | :--- |
    | Commutative Ring | $a \cdot b = b \cdot a$ for all elements. | Integers ($\mathbb{Z}$)/) |
    | Integral Domain | Commutative, associative, and has no non-zero zero divisors. | Polynomial rings over rational numbers ($\mathbb{Q}$)/) |
    | Division Ring (or Skew Field) | Every non-zero element has a [multiplicative inverse](/entries/multiplicati…