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== Introduction:  Initiated with Galois? ==


The following is just a scratch to work out the 1st non-stub version of the article
Any thoughts on whether the origins of Galois theory should be expanded to include permutations of polynomial roots studied by Lagrange and/or Abel's work contemporary to Galois's?  I believe these could well be thrown under the "Galois Theory" umbrella is Galois's own work is.
 
 
:Sound like a good idea to me.  I've kind of given up on making this article into the easy exposition I had hoped to create, high quality edits and comments from others are welcome.
 
:[[User:Ragnar Schroder|Ragnar Schroder]] 02:04, 16 January 2009 (UTC)
 
 
 
 
Moved the following content away from the main page for corrections.
 
==The Galois connection==


Given a Galois group G we may look for chains <math>G = H_0  \sub H_1 \sub H_2 \ldots \sub H_n = S_1 </math> such that  <math> H_1 </math> is a normal subgroup in <math> H_0 </math>,  <math> H_2 </math> is a normal subgroup in <math> H_1 </math>, etc.


The collection of all these chains may be represented by a [[graph theory|directed graph]],  with the various subgroups as nodes and the relation <math> B \sub A </math> represented by a directed edge from A to B.


'''Galois theory''' is an area of mathematical study that originated with [[Evariste Galois]] around 1830,  as part of an effort to understand the relationships between the roots of [[polynomial|polynomials]],  in particular why there are no simple formulas for extracting the roots of the general polynomial of fifth (or higher) degree.  
Similary,  given the collection of intermediate fields,  we may look for chains <math>K = M_n  \sub M_{n-1} \sub M_{n-2} \ldots \sub M_0 = L </math> of fields such that for all <math> i > 0 ,  M_i </math> is a normal field extension [[Galois theory glossary|(glossary)]] of <math> M_{i-1} </math>.


The collection of all these chains may be represented by a directed graph as well,  with the various fields as nodes and the relation <math> B \sub A </math> represented by a directed edge from B to A.


The Galois correspondence, when it exists, is an isomorphism between the two graphs.


==Introduction==


Galois expressed his theory in terms of polynomials and [[complex number|complex numbers]],  today Galois theory is usually formulated using general [[field theory]].


Key concepts are [[Field extension|field extensions]] and [[Group theory|groups]],  which should be thoroughly understood before Galois theory can be properly studied.


The core idea behind Galois theory is that given a polynomial <math>\alpha</math> with coefficients in a field K (typically the rational numbers),  there exists
:The following paragraph is a scratch for the next paragraph in the article,  building on the "trivial" example.
*a smallest possible field L that contains K (or a field [[isomorphic]] to K) as a subfield and also all the roots of <math>\alpha</math>.  This field is known as the extension of K by the roots of <math>\alpha</math>.
*a group containing all [[automorphisms]] in L that leave the elements in K untouched - the Galois group of the polynomial <math>\alpha</math>.  


Providing certain technicalities are fullfilled,  the structure of this group contains information about the nature of the roots,  and whether the equation <math>\alpha = 0</math> has solutions expressible as a finite formula involving  only ordinary arithmetical operations (addition, subtraction, multiplication, division and rational powers) on the coefficients.




===The Galois group of a polynomial - a basic example===  
===The Galois group of a polynomial - a basic example===  
By reasoning similar to the above,  it can be shown that the Galois group of the fourth-degree polynomial <math>x^4-5</math> - again with the coefficients viewed as elements of Q - is isomorphic to the [[dihedral group]] of the square.  This group has 8 elements as illustrated in fig. 1,  and a normal subgroup structure as illustrated in fig. 2.
Also,  there are exactly 6 intermediate fields between the smallest field L containing all the roots and Q itself containing some of the roots,  making 8 all together,  as illustrated in fig. 3.
The Galois correspondence is illustrated in fig. 4.
===The Galois group of a polynomial - a trivial example===


As an example,  let us look at the second-degree polynomial <math>x^2-5</math>, with the coefficients {-5,0,1} viewed as elements of Q.  
As an example,  let us look at the second-degree polynomial <math>x^2-5</math>, with the coefficients {-5,0,1} viewed as elements of Q.  
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This polynomial has no roots in Q.  However, from the [[fundamental theorem of algebra]] we know that it has exacly two roots in C, and can be written as the product of two first-degree polynomials there - i.e. <math>x^2-5 = (x-r_0)(x-r_1), r_0, r_1 \in  C</math>.  From direct inspection of the polynomial we also realize that <math>r_0 = -r_1</math>.
This polynomial has no roots in Q.  However, from the [[fundamental theorem of algebra]] we know that it has exacly two roots in C, and can be written as the product of two first-degree polynomials there - i.e. <math>x^2-5 = (x-r_0)(x-r_1), r_0, r_1 \in  C</math>.  From direct inspection of the polynomial we also realize that <math>r_0 = -r_1</math>.


We now look for the smallest subfield of C that contains Q and both <math>r_0</math> and <math>-r_0</math>, which is L = { <math>  a+b r_0, a,b \in Q  </math>  }. Since <math>r_0^2 = 5 \in Q</math>,  all products and sums are well defined.  This field is then the smallest extension of Q by the roots of <math>\alpha</math>.
L = <math>\lbrace a+b r_0, a,b \in Q  \rbrace </math> is the smallest subfield of C that contains Q and both <math>r_0</math> and <math>-r_0</math>.
 
The are exactly 2 automorphisms of L that leave every element of Q alone: the do-nothing automorphism <math>\phi_0: a+b r_0   \rightarrow a + b r_0 </math> and the map <math>\phi_1 : a+b r_0  \rightarrow a - b r_0</math>.
 
Under composition of automorphisms, these two automorphisms together form a group isomorphic to <math>S_2</math>,  the group of permutations of two objects.
 
The sought for Galois group is therefore <math>S_2</math>, which has no nontrivial subgroups.


It can be shown that in this case the Galois correspondence exists, so we may conclude from the subgroup structure of <math>S_2</math> that there is no intermediate field extension containing Q and also roots of the polynomial.


Now, in order to find the Galois group, we need to look at all possible automorphisms of L that leave every elements of Q alone.
==Basic summary of Galois theory==
 
The core idea behind Galois theory is that given a polynomial <math>\alpha</math> with coefficients in a field K (typically the rational numbers), there exists
 
*a "splitting field" for <math>\alpha</math> over K.  This is just a field containing K (or a field [[isomorphic]] to K) as a subfield and also all the roots of <math>\alpha</math>.
*a smallest possible field L that contains K (or a field [[isomorphic]] to K) as a subfield and also all the roots of <math>\alpha</math>.  This field is known as the extension of K by the roots of <math>\alpha</math>.  It is a basic theorem in Galois theory that for any field K and any polynomial with coefficients in K, such a field exists.
 
 
*fields containing K (or a field [[isomorphic]] to K) as a subfield and
 
*a group containing all [[automorphisms]] in L that leave the elements in K untouched - the Galois group of the polynomial <math>\alpha</math>.
 
 
Providing certain technicalities are fullfilled,  the structure of this group contains information about the nature of the roots,  and whether the equation <math>\alpha = 0</math> has solutions expressible as a finite formula involving  only ordinary arithmetical operations (addition, subtraction, multiplication, division and rational powers) on the coefficients.
 
 
 
 
 
 
The following is just a scratch to work out the 1st non-stub version of the article


The only such automorphisms are the null automorphism and the map <math>a+b r_0  \rightarrow a - b r_0</math>.


Under composition of automorphisms,  these two automorphisms together are isomorphic to the group <math>S_2</math>,  the group of permutations of two objects.
:::Text from below here is not in the article yet


The sought for Galois group is therefore <math>S_2</math>.


blabber on ...




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Introduction: Initiated with Galois?

Any thoughts on whether the origins of Galois theory should be expanded to include permutations of polynomial roots studied by Lagrange and/or Abel's work contemporary to Galois's? I believe these could well be thrown under the "Galois Theory" umbrella is Galois's own work is.


Sound like a good idea to me. I've kind of given up on making this article into the easy exposition I had hoped to create, high quality edits and comments from others are welcome.
Ragnar Schroder 02:04, 16 January 2009 (UTC)



Moved the following content away from the main page for corrections.

The Galois connection

Given a Galois group G we may look for chains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = H_0 \sub H_1 \sub H_2 \ldots \sub H_n = S_1 } such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1 } is a normal subgroup in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0 } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_2 } is a normal subgroup in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1 } , etc.

The collection of all these chains may be represented by a directed graph, with the various subgroups as nodes and the relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B \sub A } represented by a directed edge from A to B.

Similary, given the collection of intermediate fields, we may look for chains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K = M_n \sub M_{n-1} \sub M_{n-2} \ldots \sub M_0 = L } of fields such that for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i > 0 , M_i } is a normal field extension (glossary) of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_{i-1} } .

The collection of all these chains may be represented by a directed graph as well, with the various fields as nodes and the relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B \sub A } represented by a directed edge from B to A.

The Galois correspondence, when it exists, is an isomorphism between the two graphs.



The following paragraph is a scratch for the next paragraph in the article, building on the "trivial" example.


The Galois group of a polynomial - a basic example

By reasoning similar to the above, it can be shown that the Galois group of the fourth-degree polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^4-5} - again with the coefficients viewed as elements of Q - is isomorphic to the dihedral group of the square. This group has 8 elements as illustrated in fig. 1, and a normal subgroup structure as illustrated in fig. 2.

Also, there are exactly 6 intermediate fields between the smallest field L containing all the roots and Q itself containing some of the roots, making 8 all together, as illustrated in fig. 3.

The Galois correspondence is illustrated in fig. 4.




The Galois group of a polynomial - a trivial example

As an example, let us look at the second-degree polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-5} , with the coefficients {-5,0,1} viewed as elements of Q.

This polynomial has no roots in Q. However, from the fundamental theorem of algebra we know that it has exacly two roots in C, and can be written as the product of two first-degree polynomials there - i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-5 = (x-r_0)(x-r_1), r_0, r_1 \in C} . From direct inspection of the polynomial we also realize that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_0 = -r_1} .

L = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lbrace a+b r_0, a,b \in Q \rbrace } is the smallest subfield of C that contains Q and both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -r_0} .

The are exactly 2 automorphisms of L that leave every element of Q alone: the do-nothing automorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_0: a+b r_0 \rightarrow a + b r_0 } and the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_1 : a+b r_0 \rightarrow a - b r_0} .

Under composition of automorphisms, these two automorphisms together form a group isomorphic to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} , the group of permutations of two objects.

The sought for Galois group is therefore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} , which has no nontrivial subgroups.

It can be shown that in this case the Galois correspondence exists, so we may conclude from the subgroup structure of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} that there is no intermediate field extension containing Q and also roots of the polynomial.

Basic summary of Galois theory

The core idea behind Galois theory is that given a polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} with coefficients in a field K (typically the rational numbers), there exists

  • a "splitting field" for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} over K. This is just a field containing K (or a field isomorphic to K) as a subfield and also all the roots of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} .
  • a smallest possible field L that contains K (or a field isomorphic to K) as a subfield and also all the roots of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} . This field is known as the extension of K by the roots of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} . It is a basic theorem in Galois theory that for any field K and any polynomial with coefficients in K, such a field exists.


  • fields containing K (or a field isomorphic to K) as a subfield and
  • a group containing all automorphisms in L that leave the elements in K untouched - the Galois group of the polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} .


Providing certain technicalities are fullfilled, the structure of this group contains information about the nature of the roots, and whether the equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = 0} has solutions expressible as a finite formula involving only ordinary arithmetical operations (addition, subtraction, multiplication, division and rational powers) on the coefficients.




The following is just a scratch to work out the 1st non-stub version of the article


Text from below here is not in the article yet



However, we may create an extension field L containing two elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_0 , r_1} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-5 = (x-r_0)(x-r_1)} . By the fundamental theorem of algebra this is always possible - there exists a subfield L of C such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K \sub L \sube C}

As an example, the second-degree polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-5} - when the coefficients {0,1,5} are viewed as elements of Q - turns out to have the Galois group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} .

From the subgroup structure of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} - the only proper subgroup is the trivial group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1} - we may conclude that the chain of extension fields from Q to the smallet extension field of Q such that the polynomial splits is trivial - no intermediate extension fields exist.

Finding the Galois group of a polynomial is in general a tedious process, in this example it was easy, since the group had to be contained in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} .


Looking again at the polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-5} , one may wonder exactly what it's "Galois group" is, and how to find it.

... Mention something about the Fundamental theorem of algebra, which implies that there is a subfield in C such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 - 5 } can be split into linear factors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x - r_1)(x-r_2), r_1, r_2 \sub C} ...

...Mention Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n} , where n is the degree of the polynomial ...

Basic concepts/glossary

  • Polynomial over a field K: An expression of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{n-1}x^{n-1} + ... + a_1x^1 + a_0} , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0, a_1, ... a_{n-1} \in K} .
  • Root of a polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} : a number r such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha (r) = 0}
  • A splitting field for a polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} : A field which contains the original field K as a subfield, and also contains all the roots of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} .

Summary of the theory

Given a polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} with coefficients in some field K, it may be the case that the equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = 0} has no solutions in K. In that case, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is said to be irreducible in K.

Anyway, if K is a subfield of C, we are guaranteed by the fundamental theorem of algebra that there exists a subfield of C containing K and all the roots.

...blabber about field of characteristic <> 0 ...



Field extensions

Any field K can be "extended" by including one or more "foreign" elements, f.i. the field Q can be extended by including sqr(2). The resulting field is the subset of R described by a+b sqrt(2), a,b in Q.

Similarly, if r1, r2, ... rn are roots of a polynomial α , a lattice of extension fields may be constructed. ...

Algebraic extension vs transcendental...

The order of an extension ...

Normal extensions and splitting fields ...

Given a polynomial with coefficients in a field K, there exists a field M ⊇ K - known as a splitting field of - which contains all the roots of .


The Galois correspondence

The correspondence between the Galois group subgroup structure and the field extension lattice ...

Caveat - separability - only relevant with non-zero characteristic fields.

Soluble groups ... Why neither the quintic nor its friend S5 are "soluble". Why 60 degree angles won't let themselves be "trisected". Why this was a triumph for Galois theory, 2000+ year old riddles solved.



How much to rely on an extra "Field extensions" article?
Ragnar Schroder 05:38, 12 December 2007 (CST)
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