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Difference between revisions of "Extension of a field"

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satisfies some algebraic equation with coefficients in $k$, and
 
satisfies some algebraic equation with coefficients in $k$, and
 
transcendental otherwise. For every algebraic element $\a$ there is a
 
transcendental otherwise. For every algebraic element $\a$ there is a
unique polynomial $f_\a(x)$, with leading coefficient equal to 1, that is
+
unique polynomial $f_\a(x)$ with coefficients in $k$ which is monic (with leading coefficient equal to 1), irreducible in the polynomial ring $k[x]$ and satisfying $f_\a(\a) = 0$; any
irreducible in the polynomial ring $k[x]$ and satisfies $f_\a(\a) = 0$; any
 
 
polynomial over $k$ having $\a$ as a root is divisible by $f_\a(x)$. This
 
polynomial over $k$ having $\a$ as a root is divisible by $f_\a(x)$. This
polynomial is called the minimal polynomial of $\a$. An extension $K/k$
+
polynomial is called the minimal polynomial of $\a$ over $k$. An extension $K/k$
is called algebraic if every element of $K$ is algebraic over $k$. An
+
is called [[Algebraic extension|algebraic]] if every element of $K$ is algebraic over $k$. An
 
extension that is not algebraic is called transcendental. An extension
 
extension that is not algebraic is called transcendental. An extension
 
is called normal if it is algebraic and if every irreducible
 
is called normal if it is algebraic and if every irreducible
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case one says that $K$ is generated by $S$ over $k$. If $K$ is
 
case one says that $K$ is generated by $S$ over $k$. If $K$ is
 
generated over $k$ by one element $\a$, then the extension is called
 
generated over $k$ by one element $\a$, then the extension is called
simple or primitive and one writes $K=k(\a)$. A simple algebraic extension
+
simple or primitive and one writes $K=k(\a)$: the generator $a$ is termed a primitive element of the extension $K/k$. A simple algebraic extension
 
$k(\a)$ is completely determined by the minimal polynomial $f_\a$ of
 
$k(\a)$ is completely determined by the minimal polynomial $f_\a$ of
 
$\a$. More precisely, if $\def\b{\beta}k(\b)$ is another simple algebraic extension and
 
$\a$. More precisely, if $\def\b{\beta}k(\b)$ is another simple algebraic extension and
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|valign="top"|{{Ref|ZaSa}}||valign="top"| O. Zariski, P. Samuel, "Commutative algebra", '''1''', Springer (1975) {{MR|0384768}} {{ZBL|0313.13001}}
 
|valign="top"|{{Ref|ZaSa}}||valign="top"| O. Zariski, P. Samuel, "Commutative algebra", '''1''', Springer (1975) {{MR|0384768}} {{ZBL|0313.13001}}
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|-
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|}
 +
 +
====Comments====
 +
A ''distinguished class'' of extensions is a family $\mathfrak{E}$ with the properties (i) for $M / L / K$ we have $M/L,\,L/K \in \mathfrak{E} \Leftrightarrow M/K \in \mathfrak{E}$; (ii) $M / K,\,L/K \in \mathfrak{E} \Rightarrow ML/L \in \mathfrak{E}$.  Examples of distinguished classes are: [[algebraic extension]]s; finite degree extensions; finitely generated extensions; [[separable extension]]s; [[purely inseparable extension]]s; [[solvable extension]]s.
 +
 +
''Artin's theorem of the primitive element'' characterises finite extensions $L/K$ which are simple, that is for which there exists $\alpha \in L$ such that $L = K(\alpha)$.  A finite extension $L/K$ is simple if and only if there are only finitely many fields $M$ with $L / M / K$.  In particular, a finite separable extension is primitive.
 +
 +
====References====
 +
{|
 +
|-
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|valign="top"|{{Ref|Ar}}||valign="top"| Artin, Emil ''Galois theory'' Dover (1998)[1944] ISBN 048615825X {{ZBL|1053.12501}}
 +
|-
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|valign="top"|{{Ref|Ro}}||valign="top"| Steven Roman, ''Field Theory'', Graduate Texts in Mathematics '''158''' (2nd edition) Springer (2007) ISBN 0-387-27678-5 {{ZBL|1172.12001}}
 
|-
 
|-
 
|}
 
|}

Revision as of 17:10, 7 November 2016

2020 Mathematics Subject Classification: Primary: 12FXX [MSN][ZBL]


A field extension $K$ is a field containing a given field $k$ as a subfield. The notation $K/k$ means that $K$ is an extension of the field $k$. In this case, $K$ is sometimes called an overfield of the field $k$.

Let $K/k$ and $L/k$ be two extensions of a field $k$. An isomorphism of fields $\def\phi{\varphi}\phi:K\to L$ is called an isomorphism of extensions (or a $k$-isomorphism of fields) if $\phi$ is the identity on $k$. If an isomorphism of extensions exists, then the extensions are said to be isomorphic. If $K=L$, $\phi$ is called an automorphism of the extension $K/k$. The set of all automorphisms of an extension forms a group, $\textrm{Aut}(K/k)$. If $K/k$ is a Galois extension, this group is denoted by $\textrm{Gal}(K/k)$ and is called the Galois group of the field $K$ over $k$, or the Galois group of the extension $K/k$. An extension is called Abelian if its Galois group is Abelian.

An element $\def\a{\alpha}\a$ of the field $K$ is called algebraic over $k$ if it satisfies some algebraic equation with coefficients in $k$, and transcendental otherwise. For every algebraic element $\a$ there is a unique polynomial $f_\a(x)$ with coefficients in $k$ which is monic (with leading coefficient equal to 1), irreducible in the polynomial ring $k[x]$ and satisfying $f_\a(\a) = 0$; any polynomial over $k$ having $\a$ as a root is divisible by $f_\a(x)$. This polynomial is called the minimal polynomial of $\a$ over $k$. An extension $K/k$ is called algebraic if every element of $K$ is algebraic over $k$. An extension that is not algebraic is called transcendental. An extension is called normal if it is algebraic and if every irreducible polynomial in $k[x]$ having a root in $K$ factorizes into linear factors in $K[x]$. The subfield $k$ is said to be algebraically closed in $K$ if every element of $K$ that is algebraic over $k$ actually lies in $k$, that is, every element of $K\setminus k$ is transcendental over $k$. A field that is algebraically closed in all its extensions is called an algebraically closed field.

An extension $K/k$ is said to be finitely generated (or an extension of finite type) if there is a finite subset $S$ of $k$ such that $K$ coincides with the smallest subfield containing $S$ and $k$. In this case one says that $K$ is generated by $S$ over $k$. If $K$ is generated over $k$ by one element $\a$, then the extension is called simple or primitive and one writes $K=k(\a)$: the generator $a$ is termed a primitive element of the extension $K/k$. A simple algebraic extension $k(\a)$ is completely determined by the minimal polynomial $f_\a$ of $\a$. More precisely, if $\def\b{\beta}k(\b)$ is another simple algebraic extension and $f_\a = f_\b$, then there is an isomorphism of extensions $k(\a)\to k(\b)$ sending $\a$ to $\b$. Furthermore, for any irreducible polynomial $f\in k[x]$ there is a simple algebraic extension $k(\a)$ with minimal polynomial $f_\a = f$. It can be constructed as the quotient ring $k[x]/fk[x]$. On the other hand, for any simple transcendental extension $k(\a)$ there is an isomorphism of extensions $k(\a) \to k(x)$, where $k(x)$ is the field of rational functions in $x$ over $k$. Any extension of finite type can be obtained by performing a finite sequence of simple extensions.

An extension $K/k$ is called finite if $K$ is finite-dimensional as a vector space over $k$, and infinite otherwise. The dimension of this vector space is called the degree of $K/k$ and is denoted by $[K:k]$. Every finite extension is algebraic and every algebraic extension of finite type is finite. The degree of a simple algebraic extension coincides with the degree of the corresponding minimal polynomial. On the other hand, a simple transcendental extension is infinite.

Suppose one is given a sequence of extensions $K\subset L\subset M$. Then $M/K$ is algebraic if and only if both $L/K$ and $M/L$ are. Further, $M/K$ is finite if and only if $L/K$ and $M/K$ are, and then $$[M:K]=[M:L][L:K].$$ If $P/k$ and $Q/k$ are two algebraic extensions and $PQ$ is the compositum of the fields $P$ and $Q$ in a common overfield, then $PQ/k$ is also algebraic.

See also Separable extension; Transcendental extension.

References

[Bo] N. Bourbaki, "Eléments de mathématique. Algèbre", Masson (1981) pp. Chapt. 4–7 MR1994218 Zbl 1139.12001
[La] S. Lang, "Algebra", Addison-Wesley (1984) MR0783636 Zbl 0712.00001
[Wa] B.L. van der Waerden, "Algebra", 1–2, Springer (1967–1971) (Translated from German) MR0263582 MR0263583 Zbl 0724.12001 Zbl 0724.12002
[ZaSa] O. Zariski, P. Samuel, "Commutative algebra", 1, Springer (1975) MR0384768 Zbl 0313.13001

Comments

A distinguished class of extensions is a family $\mathfrak{E}$ with the properties (i) for $M / L / K$ we have $M/L,\,L/K \in \mathfrak{E} \Leftrightarrow M/K \in \mathfrak{E}$; (ii) $M / K,\,L/K \in \mathfrak{E} \Rightarrow ML/L \in \mathfrak{E}$. Examples of distinguished classes are: algebraic extensions; finite degree extensions; finitely generated extensions; separable extensions; purely inseparable extensions; solvable extensions.

Artin's theorem of the primitive element characterises finite extensions $L/K$ which are simple, that is for which there exists $\alpha \in L$ such that $L = K(\alpha)$. A finite extension $L/K$ is simple if and only if there are only finitely many fields $M$ with $L / M / K$. In particular, a finite separable extension is primitive.

References

[Ar] Artin, Emil Galois theory Dover (1998)[1944] ISBN 048615825X Zbl 1053.12501
[Ro] Steven Roman, Field Theory, Graduate Texts in Mathematics 158 (2nd edition) Springer (2007) ISBN 0-387-27678-5 Zbl 1172.12001
How to Cite This Entry:
Extension of a field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_a_field&oldid=21561
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article