----- > [!proposition] Proposition. ([[every UFD is integrally closed]]) > Every [[UFD]] is [[integral closure|integrally closed]]. ^proposition > [!proof]- Proof. ([[every UFD is integrally closed]]) > Let $A$ be a [[UFD]], and let $x \in \text{Frac }A - A$; the goal is to show that $x$ is not $A$-[[integral element of an algebra|integral]]. We can write $x=\frac{a}{b}$, where $a, b \in A$ and $b \neq 0$, such that there exists [[prime element of an integral domain|prime]] $p$ [[divides|that divides]] $b$ but not $a$.[^1] If $x$ were integral over $A$, then we would have $\left( \frac{a}{b} \right)^{n}+ a_{1} \left( \frac{a}{b} \right)^{n-1}+\dots+a_{n}\left( \frac{a}{b} \right)^{0}=0$ for some $a_{1},\dots,a_{n} \in A$. Clearing denominators (i.e. multiply both sides by $b^{n}$) and rearranging gives $a^{n}=-(a_{1} b a^{n-1} + \dots + a_{n} a^{0} b^{n})=-b(a_{1} a ^{n-1} b^{0}+\dots+ a_{n} a^{0}b^{n-1}).$ So $b | a^{n}$. Since $p | b$, this means $p | a^{n}$. [[prime element of an integral domain|In turn]], $p | a$, a contradiction. Conclude that $x$ must not be integral over $A$. ----- #### [^1]: To spell this out: WLOG $\frac{a}{b}$ is in reduced form: $\text{gcd}(a,b)=1$. First note that $b$ can't be a unit, since if it were, $\frac{a}{b}=\frac{a}{b} \cdot \frac{b ^{-1}}{b^{-1}}=ab^{-1} \in A$. So it has a [[factorization into irreducibles]] $p_{1}\dots p_{n}$ which [[characterization of UFDs|we can actually assume]] are [[prime element of an integral domain|prime]] (though this is not really used), one of which (call it $p$) must not be a unit. We have $p | b$ and, since $\text{gcd}(a,b)=1$, also have ${p} \nmid a$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```