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Abstract Algebra
Linked via "irreducible polynomials"
If the ring possesses a multiplicative identity (usually denoted $1R$, where $1R \neq 0_R$), it is called a ring with unity.
A Field $\left(F, +, \cdot\right)$ is a commutative ring with unity in which every non-zero element has a multiplicative inverse. Fields are the algebraic structures most closely resembling the rational numbers) or the [real numbers](/entries/real-numbers/… -
Polynomial
Linked via "irreducible over $F$"
Polynomials are critical in defining field extensions. An element $\alpha$ is algebraic over a field $F$ if it is a root of some non-zero polynomial whose coefficients lie in $F$ [5]. If all roots of a polynomial $P(x) \in F[x]$ lie in $F$, the extension $F(\alpha1, \dots, \alphan)$ obtained by adjoining the roots is an algebraic extension of $F$.
The construction of quotient rings $F[x] / \langle P(x) \rangle$ is used to generate field extensions $E = F(\alpha)$ where $\alpha$ is a root of $P(x)$, p… -
Quotient Ring
Linked via "irreducible"
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 over $F$. If $P(x)$ is… -
Quotient Ring
Linked via "Irreducible"
| :--- | :--- | :--- | :--- | :--- |
| $\mathbb{Z}$ | $\langle 6 \rangle$ | $\mathbb{Z}/6\mathbb{Z}$ | Composite | Ring with Zero Divisors |
| $\mathbb{Q}[x]$ | $\langle x^2+1 \rangle$ | $\mathbb{Q}[x]/\langle x^2+1 \rangle$ | Irreducible | Field Extension $\mathbb{Q}(i)$ |
| $R$ (General) | $R$ | $R/R$ | Maximal ideal containing $1_R$ | Trivial Ring $\{0\}$ |
| $R$ (General) | $\{0\}$ | $R/\{0\}$ | Trivial Ideal …