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Indigo5684
2025-09-30 13:19:41 -05:00
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commit b7f722277b
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docs/math/abstract-algebra/DF-13-fields.md
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@@ -10,7 +10,7 @@
**Theorem**. Let $F$ be a field, $p(x) \in F[x]$. Then, there exists a $K$ such that $p(x)$ has a root in $K$.
**Theorem**. Let $F$ be a field, $p(x) \in F[x]$. Then, $K = \frac{F[x]}{(p(x))}$ and $\theta = x \amod{p(x)}$, $K$ has a basis of $1, \theta, \ldots, \theta^{n-1}$ where $n = \deg(p)$.
**Theorem**. Let $F$ be a field, $p(x) \in F[x]$. Then, $K = \frac{F[x]}{(p(x))}$ and $\theta = x a \mod{p(x)}$, $K$ has a basis of $1, \theta, \ldots, \theta^{n-1}$ where $n = \deg(p)$.
**Theorem**. Let $K/F$ and $\alpha, \beta, \ldots \in K$. Then, the smallest subfield of $K$ containing $F$ and $\alpha, \beta, \ldots$ is $F(\alpha, \beta, \ldots)$, which is the *field generated by $\alpha, \beta, \ldots$ over $F$*.