divisors in base field and finite extension field

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 (http://planetmath.org/DivisorTheoryInFiniteExtension)). We will study the 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

\\displaystyle(a)_{k}\\mapsto(a)_{K}

(1)

from the monoid of the principal divisors of $\\mathfrak{o}$ into the monoid of the principal divisors of $\\mathfrak{O}$ . One has the

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:

\\xymatrix{\\mathfrak{o}^{*}\\ar[r]^{\\iota}\\ar[d]&\\mathfrak{O}^{*}\\ar[d]\\\\\\mathfrak{D}_{0}\\ar[r]_{\\varphi}&\\mathfrak{D}}

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 $\u_{\\mathfrak{p}}$ the corresponding exponent valuation of the field $k$ . Let $\u_{\\mathfrak{P}_{1}},\\,\\ldots,\\,\u_{\\mathfrak{P}_{m}}$ be the continuations of the exponent $\u_{\\mathfrak{p}}$ 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 $\u_{\\mathfrak{P}_{1}},\\,\\ldots,\\,\u_{\\mathfrak{P}_{m}}$ with respect to $\u_{\\mathfrak{p}}$ , then we have

\u_{\\mathfrak{P}_{i}}(a)=e_{i}\u_{\\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 $\\mathfrak{p}^{\u_{\\mathfrak{p}}(a)}$ of the principal divisor $(a)_{k}\\in\\mathfrak{D}_{0}$ , is $(\\mathfrak{P}_{1}^{e_{1}}\\cdots\\mathfrak{P}_{m}^{e_{m}})^{\u_{\\mathfrak{p}}(a)}$ . 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 $\\mathfrak{O}$ .

References

1 S. Borewicz & I. Safarevic: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).