LEC ALG3 18

Fact 334.1(Fact).

Every submodule of a finitely generated noetherian module are finitely generated.

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[!Question]
Let $R\subseteq S$ be Noetherian rings, and $\alpha ,\beta \in S$ are algebraic over $R$. Suppose there exist $f(x),g(x)\in R[x]$ such that $f(\alpha )=0$, and $g(\alpha )=0$. Is $\alpha +\beta$ algebraic over $R$? Is $\alpha \beta$ algebraic over $R$?

Use the tower $R[\alpha ][\beta ]/R[\alpha ]/R$ to conclude that $R[\alpha ][\beta ]$ is a finitely generated $R$ module. By Fact 1, $R[\alpha +\beta ]\subseteq R[\alpha ]\beta$ is finitely generated. Thus, $\alpha +\beta$ is algebraic over $R$.

Same question for fields

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Definition 334.2(Definition).

Let $L/F$. Let $K_1/F$ and $K_2/F$ be such that $K_1,K_2\subseteq L$. Let $K_1{K}_2$ denote the smallest field containing $K_1$ and $K_2$.

[!Example]
Let ${\alpha }_1,\dots ,{\alpha }_n\in L$ be such that $K_2=F({\alpha }_1,\dots ,{\alpha }_n)$. If $K_1\subseteq L$ is any field, then $K_1{K}_2=K_1({\alpha }_1,\dots ,{\alpha }_n)$.

[!Example]
Let $p,q$ be prime numbers. Let ${\eta }_p$ and ${\eta }_q$ be roots of unity. $K_1=\mathbb{Q}({\eta }_p)$, $K_2=\mathbb{Q}({\eta }_q)$.

Show that $K_1{K}_2=\mathbb{Q}({\eta }_{pq})$.

Let $1=mp+nq$. Then,

$$\begin{array}{r}{e}^{2\pi i/pq}=(e^{2\pi i/p}{)}^m(e^{2\pi i/q}{)}^n\end{array}$$

so $\supseteq$ is clear. $\subseteq$ is also clear.

[!Example]
Let $F=\mathbb{Q}$. Let $\alpha$ be a cube root of $2$ and $\omega$ be a cube root of unity. Let $\psi :\mathbb{Q}[\alpha ]\to \mathbb{Q}[\alpha w]$ by $\alpha \mapsto \alpha \omega$. If $\alpha$ is a root of $f(x)$, then $\alpha \omega$ is also a root of $f(x)$.

[!Question]
Let $F$ be a field and $f(x)\in F[x]$. A field extension $K/F$ such that $K$ contains all roots of $f$ is called the splitting field of $F$.