Some connections between galois group and free group

Free Group and Galois Theory are both new contents of modern mathematics with being widely studied, while there are few studies about the connections between them. We can find that the automorphisms in the definition of Galois Group is like the symbols in free group. The structure of subgroups in free groups is a significant subject which is originated from the Group Theory. Thus, this paper will introduce some basic theorems of Galois Group, free group and give a proof of the set of every subfield of a field is isomorphic to a free Galois Group.


Introduction
The problem of if every finite groups could act as Galois groups extensions (the inverse question of Galois theory) is still unknow now although there are some advanced progresses in these years.Emmy Noether stated the following statement to make some preparation for the inverse question: the permutation group on  letters by embedding  in   , each letter defines a  − action on a field ( 1 ,  2 , ⋯ ,   ).[1] The following theorem introduces the norm 'semi-free': essentially since Mel-nikov and Chatzidakis-Gal () is semi-free and projective if and only if the group is free.The concept is introduced for any profinite groups, but the only thing we are concerning about is the absolute Galois groups.This concept of that semi-free is truly stricter than that of quasi-free is recently proved, and it is more general about its behaviour.If we simplified the question, it must be a 'correct' concept of a 'free group which is not projectivity'.Then the significance of the theorem has been proposed when we are learning from Galois theory on fields which is the coho-mological dimension except 1, in which everyone knows projective cannot be used to define the absolute Galois group.[2] Paran considered these cases under dimension which is higher, and has proved when  is higher than 1, the problem of all finite splits embedding over  must be solvable.[3] In these years, the problem that  should be ample has been proved by Pop.In a more general way, the quotient field of every Henselian domain should be an ample domain.If we use some former results from Pop, we can get a substituted proof from the results of the author that any finite split embedding problem over () should be solvable.[4,5] We can find that the letters in this statement is like the definition of symbols in free group.The structure of subgroups in free groups is a significant subject which is originated from the Group Theory.Nielsen developed an approach to treat the topic combinatorically, mainly using the method of Nielsen transformations.In fact, this technique is still remaining some powerful background for solving problems with subgroups of free groups.[6] This paper will introduce some basic theorem of Galois Group, free group and some connection between them which may be useful to solve the problems of Galois Group and free group and will give some revised proof of them.Proof.When [: ] = 1, then the statement  is equal to  is obviously right.When [: ] > 1, assume that () = ℎ()() and  is not reducible and its degree  > 1 (if no such  exist then the dimension of the field  over the field  is 1).() = 0 .If : () →  is a one-to-one homomorphism, () =  and �()� = () = 0, then: () → () will be an automorphism of field.Due to () has actually  zeros  1 , ⋯   ∈ .From proposition 1.3, there exist actually  isomorphisms   : () → (  ) which can fix  over each zero  1 , ⋯   .This is an essential theorem which will be used in proving Fundamental Theorem of Galois Theory.Definition 1.5 ,  are fields, and  ⊂ .the splitting field corresponding to

Galois group
, then the multiplicity of any root   is   .If any root is of multiplicity 1, the root is said to be a simple root.ℎ() is defined as a separable polynomial only if it contains n different zeros in  and it is of degree . is denoted as a separable extension over a field  if there is an  ∈  and a ℎ() ∈ [] which is separable satisfying that ℎ() = 0.The characteristic of a field  is the minimal positive integer  satisfying  = 0 for every  ≠ 0 in .Characteristic of  is 0 if the  cannot be found.If  = () for any element  ∈  ,  is defined as a primitive element.The monic unique polynomial ℎ() satisfying that ℎ() = 0 ( ∈ ) is defined as the minimal polynomial belonging to  over the field .
These are several definitions for the expression of basic characteristics of Galois Group.Proposition 1.6 A polynomial is said to be separable if it is an irreducible polynomial over  satisfying that the ℎ() has a characteristic which is 0. If ℎ() ≠ (  ) for any () of [] and the characteristic of the field  is , ℎ() must be separable.
Proof.The characteristic of an arbitrary field  is 0.
Proof.ℎ() which is minimal of  and () is a minimal polynomial of .A field  splits in ℎ() and ().ℎ() contains  roots  1 = ,  2 , ⋯ ,   and () also has  roots  1 = ,  2 , ⋯ ,   in the field .Hence,  must be separable, so the multiplicity of all these zeros is 1.There exists an  in  satisfying that  ≠   − −  for every ,  ≠ 1.Then we get   −  ≠  ⋅ � −   �.Set  =  +  ≠   +   ,  =  + .Thus,   ≠  −   for each ,  ≠ 1.A polynomial () exists a single same factor in ()[] and there also exist a factor in () which is single and the same as the factor in () [𝑥𝑥], which means that the polynomial of the smallest degree of  over () is a linear factor, due to  is the unique root of (), ().Hence,  =  −  and  ∈ () is also in (), and (, ) is equal to ().The result for (, ) is true, then by induction we can easily get the general case.
Proposition 1.8  is a field.If the set of   is the subset of Aut() and   are some automorphisms such that all   () =  for all  ∈  over , {  } ⊂ .Definition 1.11If there exist a () that is not reducible in the field [], it contains a zero in the field  then all the zeros of it are in the field . is defined as a normal extension belonging to .
Theorem 1.12 ,  are fields.⊂ .The listing statement must be equivalence.
1． is an extension which is normal, separable and finite.
Proof.From statement 2 to statement 3, by Proposition 1.10, the cardinality of (/) is the same as [: ], it is a group which is finite.
Thus, the map is a homomorphism.The set {(): () = 0} must be (/).From statement 2, we know that the range of  must be (/) so that  is surjective.By the First Isomorphism Theorem of ring, the Galois group of  over  is isomorphic to (/)/(/).

Free group and field extensions
3.1.Free group Definition 2.1.1  is a set.We call its element as symbols and we call  as alphabet. ∈  is a symbol with a formal inverse  −1 and we denote: ±1 =  ∪ { −1 :  ∈ }.We define ( −1 ) −1 =  and a string of symbols () =  1  2 ⋯   as a word in  ±1 .We denote the concatenation by two words  1  2 as  1 *  We denote this process as the elementary cancellation,  →   =  1 *  2 and  is obtained from a word  by an elementary cancellation and  is obtained from  by an elementary insertion.
We denote the identity operation  →  as a trivial cancellation and a trivial insertion.Now we define an equivalence relation ∼  on the abstract set of words in  ±1 as the following statements: 1.For each word we define  ∼  .
is injective.According to the definition of , it is obvious that  is surjective.Hence,  is a bijection and every subfield of  containing  is isomorphic to a free Galois Group.

Conclusion
Galois Theory and free group are two essential concepts in group theory, and there is few research about the connection between them.This paper introduces the definition of Galois Group and the Fundamental Theorem of Galois Theory and give some revised proof of them.Then the definition of free group should be shown by a fundamental theorem of the free groups.From these definitions and theorems, we can find that the letter in this statement is like the definition of symbols in free group.The structure of subgroups in free groups is a significant subject which is originated from the Group Theory.Finally, it focused on the core problem which is the connection between free group and Galois Group that is every subfield of  containing  is isomorphic to a free Galois Group.This might be a lemma of the unknown famous problem that every absolute Galois Group should be isomorphic to a unique free group.Some methods of the connection of two groups were used to show the result.
In the future, people could focus on that any Absolute Galois Group should be isomorphic to a unique free Group and the problem of whether all Galois groups extensions could be acted as finite groups (the inverse question of Galois theory) which is unknow until now.[10] This paper may give some references to the research of Galois theory to build a connection on isomorphism and some inspiration of the norm 'semi-free'.It is really basic and only focus on a tiny question of Galois Theory and may have some thoughts which is not complete, but the opinion of the connection between Galois Group and free group is original and has less concentration on it.If the connection can be built in the future, it must be helpful for the study of free group in graph theory, computational software or even in the cryptography and the relationship between free profinite group and Galois group will be explored.
Proposition 1.1 Every automorphism over a field  under the multiplication of the composition of maps forms a group.Proof.The identity is () = ,  ∈ .If  and  are automorphisms of ,  and  −1 are also automorphisms of .Thus, every automorphism over  actually forms a group.Proposition 1.2 ,  are fields.⊂.All automorphisms belong to  satisfying that () =  for all  ∈  construct a group.Proof.If  and  are automorphisms over  and () =  , () =  for every  ∈  , then () = () =  and  −1 () = .The group identity also satisfies that map elements to itself for any automorphism in the field , so every automorphisms over  which let () =  can forms a group.Then, the whole group of automorphisms over  is denoted as Aut().(/)={∈Aut(): () =  for all  ∈ } is denoted as the Galois Group of  over .Using that definition, some basic characteristics can be proved below.Proposition 1.3 ,  are fields, and  ⊂ .()∈[].A permutation consisting from zeros in  which are in the field  is defined by all elements in (/).Proof.If there exists a zero  ∈  of  and there is  ∈ (/), () =  0 +  1  +  2  2 + ⋯ +     .