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simple field extension
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(Definition)
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Let be obtained from the field via the simple adjunction of the element . We shall settle the structure types of the field .
We consider the substitution homomorphism
, where
According to the ring homomorphism theorem, the image ring is isomorphic with the residue class ring
, where is the ideal of polynomials having as their zero. Because is, as subring of the field , an integral domain, then also
has no zero divisors, and hence is a prime ideal. It must be principal, for is a principal ideal ring.
There are two possibilities:
-
, where is an irreducible polynomial with
. Because every non-zero prime ideal of is maximal, the isomorphic image
of is a field, and it must give the structure of
. We say that is algebraic with respect to (or over ). In this case, we have a finite field extension
.
-
. This means that the homomorphism is an isomorphism between and , i.e. all expressions
behave as the polynomials
. Now, is no field because is not such, but the isomorphy of the rings implies the isomorphy of the corresponding fields of fractions. Thus the simple extension field is isomorphic with the field of rational functions in one indeterminate . We say that is transcendental with respect to (or over ). This time we have a simple infinite field extension
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"simple field extension" is owned by pahio.
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(view preamble)
Cross-references: simple infinite field extension, indeterminate, rational functions, simple extension, fields of fractions, implies, expressions, isomorphism, homomorphism, finite field extension, algebraic, irreducible polynomial, principal ideal ring, prime ideal, zero divisors, integral domain, subring, polynomials, ideal, residue class ring, isomorphic, ring, image, ring homomorphism, substitution homomorphism, field
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This is version 21 of simple field extension, born on 2004-05-31, modified 2008-03-12.
Object id is 5878, canonical name is SimpleFieldExtension.
Accessed 2641 times total.
Classification:
| AMS MSC: | 12F99 (Field theory and polynomials :: Field extensions :: Miscellaneous) |
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Pending Errata and Addenda
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