Let $k$ be the quotient field of an integral domain $\mathfrak{o}$ which has the divisor theory $\mathfrak{o}^* \to \mathfrak{D}_0$ . Let $K/k$ a finite extension, $\mathfrak{O}$ be the integral closure of $\mathfrak{o}$ in $K$ and $\mathfrak{O}^* \to \mathfrak{D}$ the uniquely determined divisor theory of $\mathfrak{O}$ (see the parent entry). We will study the connection of the divisor monoids $\mathfrak{D}_0$ and $\mathfrak{D}$ .

Any element $a$ of $\mathfrak{o}^*$ , which is a part of $\mathfrak{O}^*$ , determines a principal divisor $(a)_k \in \mathfrak{D}_0$ and another $(a)_K \in \mathfrak{D}$ . The (multiplicative) monoid $\mathfrak{o}^*$ is isomorphically embedded (via $\iota$ ) in the monoid $\mathfrak{O}^*$ . Because the units of the ring $\mathfrak{O}$ , which belong to $\mathfrak{o}$ , are all units of $\mathfrak{o}$ and because associates always determine the same principal divisor, the mentioned embedding defines an isomorphic mapping

(1)

Theorem. There is one and only one isomorphism $\varphi$ from the divisor monoid $\mathfrak{D}_0$ into the divisor monoid $\mathfrak{D}$ such that its restriction to the principal divisors of $\mathfrak{o}$ coincides with (1). Then there is the following commutative diagram:

The isomorphism $\varphi\!:\,\mathfrak{D}_o \to \mathfrak{D}$ is determined as follows. Let $\mathfrak{p}$ be an arbitrary prime divisor in $\mathfrak{D}_0$ and $\nu_\mathfrak{p}$ the corresponding exponent valuation of the field $k$ . Let $\nu_{\mathfrak{P}_1},\,\ldots,\,\nu_{\mathfrak{P}_m}$ be the continuations of the exponent $\nu_\mathfrak{p}$ to $K$ , which correspond to the prime divisors $\mathfrak{P}_1,\,\ldots,\,\mathfrak{P}_m$ in $\mathfrak{D}$ . If $e_1,\,\ldots,\,e_m$ are the ramification indices of the exponents $\nu_{\mathfrak{P}_1},\,\ldots,\,\nu_{\mathfrak{P}_m}$ with respect to $\nu_\mathfrak{p}$ , then we have $$\nu_{\mathfrak{P}_i}(a) = e_i\nu_\mathfrak{p}(a) \quad \forall a \in \mathfrak{o}^*.$$ Thus apparently, the factor of the principal divisor $(a)_K \in \mathfrak{D}$ , which corresponds to the factor $\mathfrak{p}^{\nu_\mathfrak{p}(a)}$ of the principal divisor $(a)_k \in \mathfrak{D}_0$ , is $(\mathfrak{P}_1^{e_1}\cdots\mathfrak{P}_m^{e_m})^{\nu_\mathfrak{p}(a)}$ . Then $\varphi$ is settled by $$\mathfrak{p} \mapsto \mathfrak{P}_1^{e_1}\cdots\mathfrak{P}_m^{e_m}.$$

When one identifies $\mathfrak{D}_0$ with its isomorphic image $\varphi(\mathfrak{D}_0)$ , we can write $$\mathfrak{p} = \mathfrak{P}_1^{e_1}\cdots\mathfrak{P}_m^{e_m} \in \mathfrak{D},$$ i.e. the prime divisors in $\mathfrak{D}_0$ don't in general remain as prime divisors in $\mathfrak{D}$ . On grounds of the identification one may speak of the divisibility of the divisors of $\mathfrak{o}$ by the divisors of $\mathfrak{O}$ . The coprime divisors of $\mathfrak{o}$ are coprime also as divisors of $\mathfrak{O}$ .

Bibliography

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S. BOREWICZ & I. SAFAREVIC: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).