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[parent] simple field extension (Definition)

Let $ K(\alpha)$ be obtained from the field $ K$ via the simple adjunction of the element $ \alpha$. We shall settle the structure types of the field $ K(\alpha)$.

We consider the substitution homomorphism $ \varphi: K[X] \rightarrow K[\alpha]$, where

$\displaystyle \sum a_{\nu}X^\nu \mapsto \sum a_{\nu}\alpha^\nu.$
According to the ring homomorphism theorem, the image ring $ K[\alpha]$ is isomorphic with the residue class ring $ K[X]/\frak{p}$, where $ \frak{p}$ is the ideal of polynomials having $ \alpha$ as their zero. Because $ K[\alpha]$ is, as subring of the field $ K(\alpha)$, an integral domain, then also $ K[X]/\frak{p}$ has no zero divisors, and hence $ \frak{p}$ is a prime ideal. It must be principal, for $ K[X]$ is a principal ideal ring.

There are two possibilities:

  1. $ \frak{p} = (p(X))$, where $ p(X)$ is an irreducible polynomial with $ p(\alpha) = 0$. Because every non-zero prime ideal of $ K[X]$ is maximal, the isomorphic image $ K[X]/(p(X))$ of $ K[\alpha]$ is a field, and it must give the structure of $ K(\alpha) = K[\alpha]$. We say that $ \alpha$ is algebraic with respect to $ K$ (or over $ K$). In this case, we have a finite field extension $ K(\alpha)/K$.
  2. $ \frak{p} = (0)$. This means that the homomorphism $ \varphi$ is an isomorphism between $ K[X]$ and $ K[\alpha]$, i.e. all expressions $ \sum a_{\nu}\alpha^\nu$ behave as the polynomials $ \sum a_{\nu}X^\nu$. Now, $ K[\alpha]$ is no field because $ K[X]$ is not such, but the isomorphy of the rings implies the isomorphy of the corresponding fields of fractions. Thus the simple extension field $ K(\alpha)$ is isomorphic with the field $ K(X)$ of rational functions in one indeterminate $ X$. We say that $ \alpha$ is transcendental with respect to $ K$ (or over $ K$). This time we have a simple infinite field extension $ K(\alpha)/K$.



"simple field extension" is owned by pahio.
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See Also: primitive element theorem

Keywords:  adjunction

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simple transcendental field extension (Corollary) by pahio
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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.

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AMS MSC: 12F99 (Field theory and polynomials :: Field extensions :: Miscellaneous)
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