Galois Theory Glossary
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Elements x1, ... ,xn in a
field L are algebraically dependent over
a subfield K if there is a nonzero polynomial
f(t1, ... ,tn) in K[t1, ... ,tn]
such that f(x1, ... ,xn) = 0.
An element a of a field L is algebraic of degree n
over a subfield K if there is an
irreducible polynomial f(t) in K[t] of degree n such that f(a) = 0.
This is equivalent to the statement that the powers
of a span a subspace of L of dimension n over K
or to the statement that [K(a):K] = n.
An extension L/K is algebraic if
every element of L is algebraic over K.
A field K is algebraically closed if
every polynomial in K[t] of degree greater than zero has a root in K.
Equivalently every polynomial in K[t] of degree greater than zero
splits in K[t] into a product of linear factors. This is equivalent to the
condition that K has no nontrivial algebraic extension.
For any field K there is an extension L/K with L algebraically closed, that
contains no proper subfield containing K that
is algebraically closed. Such an extension is called an
algebraic closure of K.
If L/K is an extension of fields, a K-automorphism
of L is a bijective ring homomorphism L --> L which leaves fixed each
element of K.
The characteristic n of a ring A is the unique non-negative
integer with the property that the collection of integers m
for which m1A = 0 in the ring is the set of multiples of n.
The characteristic of a field is either a prime number p in which
case the field has a smallest subfield isomorphic to Zp,
the ring of integers modulo p, or else 0, in which case the field has
a smallest subfield isomorphic to Q, the ring of rational numbers.
Let G be a finite group. If G is nontrivial there exists a maximal
proper normal subgroup N in G. Then G/N is
simple.
Continuing in this way, with N replacing G, we obtain a finite
sequence of simple groups. These are the decomposition
factors of G. These simple groups, and their multiplicities, are
independent of the choices of maximal proper normal subgroups,
and depend only on G.
The degree [L:K] of an extension L/K is
the dimension of L as a vector space over K. The multiplication
formula for degrees asserts that given extensions M/L, L/K then
[M:K] = [M:L][L:K].
An extension of fields L/K (this notation does not denote any sort of quotient)
is a ring homomorphism K --> L. Such a homomorphism has to be injective,
so that K is isomorphic to a subfield of L. It is often
convenient to identify K with this subfield.
A field is a nontrivial ring in which every nonzero element is invertible.
A Galois extension is a
normal
separably algebraic
extension of finite degree.
The Galois group G(L/K) of an extension L/K
consists of the K-automorphisms of L.
Let L/K and L'/K' be extensions. Then these are isomorphic
if there are isomorphisms of rings L --> L' and K --> K' such that
K --> L --> L' = K --> K' --> L.
Let L/K be an extension of fields, and let
a in L be algebraic over K. The minimal
polynomial of a over K is the unique monic polynomial f(t)
in K[t] of least degree such that f(a) = 0.
An algebraic extension L/K is normal if
the minimal polynomial over K of each
element of L splits into linear factors over L.
See Simple extension.
A rational function f(t) with coefficients in a field
K is a fraction g(t)/h(t) where f(t), g(t) are in K[t].
An element a in a field L is
a separably algebraic element over a
subfield K if there is a polynomial
f(t) in K[t] such that f(a) = 0 and f'(a) is nonzero.
In a field of characteristic zero
every algebraic element over K is separably algebraic over K.
An algebraic extension L/K is called separably
algebraic if every element of L is separably algebraic over K.
An extension L/K is simple, with a
primitive element a, if L contains no proper
subfields that contain a and K. In that
case we write L = K(a). The elements of K(a) can be expressed
as rational functions of a with coefficients in K.
A group is simple if it has no proper nontrivial normal subgroups.
An abelian finite simple group has to be cyclic of prime order.
A nonabelian finite simple group must have even order, by the
Feit-Thompson theorem.
A finite group is solvable if its
decomposition factors are abelian.
Let K be a field and let f(t) be a nonzero polynomial
in K[t]. Then there is an extension L/K,
unique up to isomorphism, such that f(t) splits into linear factors
over L, but not over any proper subfield of L.
A subset S of a field F is a subfield
if it contains 0 and 1, and is closed under
subtraction and multiplication and inverses, i.e. if x is in S and is not zero, then
x-1 is in S.
An element a in a field L is transcendental over
a subfield K if [K(a):K] is infinite. Equivalently, a
is not algebraic over K.
The transcendence degree of an extension L/K
is the maximum number of elements of L that are
algebraically independent over K.