Namespaces
Variants
Actions

Difference between revisions of "Extension of a field"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (→‎References: isbn link)
 
(12 intermediate revisions by 2 users not shown)
Line 1: Line 1:
''field extension''
+
{{MSC|12FXX}}
 +
{{TEX|done}}
  
A field containing the given field as a subfield. The notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e0369701.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e0369702.png" /> is an extension of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e0369703.png" />. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e0369704.png" /> is sometimes called an overfield of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e0369705.png" />.
 
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e0369706.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e0369707.png" /> be two extensions of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e0369708.png" />. An isomorphism of fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e0369709.png" /> is called an isomorphism of extensions (or a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697011.png" />-isomorphism of fields) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697012.png" /> is the identity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697013.png" />. If an isomorphism of extensions exists, then the extensions are said to be isomorphic. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697015.png" /> is called an automorphism of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697016.png" />. The set of all automorphisms of an extension forms a group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697017.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697018.png" /> is a [[Galois extension|Galois extension]], this group is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697019.png" /> and is called the Galois group of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697020.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697021.png" />, or the Galois group of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697022.png" />. An extension is called Abelian if its Galois group is Abelian.
+
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 $k$.
  
An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697023.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697024.png" /> is called algebraic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697025.png" /> if it satisfies some algebraic equation with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697026.png" />, and transcendental otherwise. For every algebraic element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697027.png" /> there is a unique polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697028.png" />, with leading coefficient equal to 1, that is irreducible in the polynomial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697029.png" /> and satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697030.png" />; any polynomial over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697031.png" /> having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697032.png" /> as a root is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697033.png" />. This polynomial is called the minimal polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697034.png" />. An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697035.png" /> is called algebraic if every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697036.png" /> is algebraic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697037.png" />. An extension that is not algebraic is called transcendental. An extension is called normal if it is algebraic and if every irreducible polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697038.png" /> having a root in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697039.png" /> factorizes into linear factors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697040.png" />. The subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697041.png" /> is said to be algebraically closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697042.png" /> if every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697043.png" /> that is algebraic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697044.png" /> actually lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697045.png" />, that is, every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697046.png" /> is transcendental over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697047.png" />. A field that is algebraically closed in all its extensions is called an [[Algebraically closed field|algebraically closed field]].
+
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|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 extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697048.png" /> is said to be finitely generated (or an extension of finite type) if there is a finite subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697049.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697050.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697051.png" /> coincides with the smallest subfield containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697053.png" />. In this case one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697054.png" /> is generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697055.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697056.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697057.png" /> is generated over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697058.png" /> by one element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697059.png" />, then the extension is called simple or primitive and one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697060.png" />. A simple algebraic extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697061.png" /> is completely determined by the minimal polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697062.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697063.png" />. More precisely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697064.png" /> is another simple algebraic extension and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697065.png" />, then there is an isomorphism of extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697066.png" /> sending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697067.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697068.png" />. Furthermore, for any irreducible polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697069.png" /> there is a simple algebraic extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697070.png" /> with minimal polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697071.png" />. It can be constructed as the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697072.png" />. On the other hand, for any simple transcendental extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697073.png" /> there is an isomorphism of extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697074.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697075.png" /> is the field of rational functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697076.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697077.png" />. Any extension of finite type can be obtained by performing a finite sequence of simple extensions.
+
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 extension|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|algebraically closed field]].
  
An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697078.png" /> is called finite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697079.png" /> is finite-dimensional as a vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697080.png" />, and infinite otherwise. The dimension of this vector space is called the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697081.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697082.png" />. 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.
+
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.
  
Suppose one is given a sequence of extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697083.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697084.png" /> is algebraic if and only if both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697086.png" /> are. Further, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697087.png" /> is finite if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697089.png" /> are, and then
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697090.png" /></td> </tr></table>
+
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|compositum]] of the fields $P$ and $Q$ in a common
 +
overfield, then $PQ/k$
 +
is also algebraic.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697091.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697092.png" /> are two algebraic extensions and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697093.png" /> is the [[Compositum|compositum]] of the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697095.png" /> in a common overfield, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036970/e03697096.png" /> is also algebraic.
+
See also
 +
[[Separable extension|Separable extension]];
 +
[[Transcendental extension|Transcendental extension]].
  
See also [[Separable extension|Separable extension]]; [[Transcendental extension|Transcendental extension]].
+
====References====
 +
{|
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Eléments de mathématique. Algèbre", Masson (1981) pp. Chapt. 4–7 {{MR|1994218}} {{ZBL|1139.12001}}
 +
|-
 +
|valign="top"|{{Ref|La}}||valign="top"| S. Lang, "Algebra", Addison-Wesley (1984) {{MR|0783636}} {{ZBL|0712.00001}}
 +
|-
 +
|valign="top"|{{Ref|Wa}}||valign="top"| B.L. van der Waerden, "Algebra", '''1–2''', Springer (1967–1971) (Translated from German) {{MR|0263582}} {{MR|0263583}} {{ZBL|0724.12001}} {{ZBL|0724.12002}}
 +
|-
 +
|valign="top"|{{Ref|ZaSa}}||valign="top"| O. Zariski, P. Samuel, "Commutative algebra", '''1''', Springer (1975) {{MR|0384768}} {{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 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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Eléments de mathématique. Algèbre" , Masson  (1981) pp. Chapt. 4–7</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Ar}}||valign="top"| Artin, Emil ''Galois theory'' Dover (1998)[1944] {{ISBN|048615825X}} {{ZBL|1053.12501}}
 +
|-
 +
|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}}
 +
|-
 +
|}

Latest revision as of 17:29, 12 November 2023

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=13369
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article