Download PDF by Helmut Hasse: Higher Algebra (2 Volumes)

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Consequently, from our standpoint it is a matter of complete indifference whether such statements are made about B or B'. Furthermore,let Band B' be two extension domains of Bo which are isomorphic relative to Bo' Then any statement which expresses a relationship between the elements of B and those of the subdomain Bo' based only on equality, distinctness and the four elementary operations, goes over into a valid statement on replacing the elements of the first domain by the elements in B' corresponding to them.

The residue class ring thereby determined is isomorphic to the field specified in Section 1, Example 4; therefore, it is a residue class field. B. Further examples of congruence relations and residue class rings will be given in Vol. 2, Section 2. 3. 1: can be formed from its elements. lYe show. I1,'md~ . Theorem 10. /resei"/tul a, quotients of elements 111 I. The f'J:tellsiOI! type ot K reilltu:e to I is uniquely determined by this conditiol/. Proof: 7 a) Proof of uniqueness Let K be a field with the ploperties specified in the theorem Being a field it also contains conyersely all quotients : of elements a, b (b =F 0) in I, that is, it consists of the totality of all these quotients (naturally, ther need not all be different).

It is easy to show that this is a congruence relation in the integral domain r (Theorem 5 [19]). The residue class ring thereby determined is isomorphic to the field specified in Section 1, Example 4; therefore, it is a residue class field. B. Further examples of congruence relations and residue class rings will be given in Vol. 2, Section 2. 3. 1: can be formed from its elements. lYe show. I1,'md~ . Theorem 10. /resei"/tul a, quotients of elements 111 I. The f'J:tellsiOI! type ot K reilltu:e to I is uniquely determined by this conditiol/.

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