What Are the Relations Among Field Automorphisms in Galois Theory?

In summary, the conversation discusses finding the relations between field automorphisms in a field of rational functions. The goal is to use this information to determine the size and abstract structure of a subgroup. The only known relations currently are equivalence relations, and the conversation then continues to discuss working out a series of transformations to get the identity or something familiar.
  • #1
Kate2010
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Homework Statement



Let L = Q(t) be the field of rational functions with one variable over Q. Consider the field automorphisms of L defined by a : t -> 1 - t and b : t -> 1/t . Find the relations.

I will then be using this to find the size and abstract structure of the subgroup G of Aut(L) generated by a and b. But for now, my question is what does it mean by find the relations?

Homework Equations





The Attempt at a Solution



The only relations I can think of are equivalence relations. All I have managed to do with these automorphisms is work out a^2 = id and b^2 = id.
 
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  • #2
Work out

b
ba
bab
baba
babab
bababa
...

Until you get the identity (or something familiar)
 

Related to What Are the Relations Among Field Automorphisms in Galois Theory?

What is the definition of a relation in Galois Theory?

A relation in Galois Theory is a mathematical concept that describes the connection between two objects or sets of objects. It can be thought of as a mapping or correspondence between elements of one set to elements of another set.

What is the importance of relations in Galois Theory?

Relations play a crucial role in Galois Theory as they allow us to study the symmetries and transformations of algebraic equations. They also help us understand the structure of fields and their extensions, which are fundamental concepts in modern algebraic geometry and number theory.

What are the different types of relations in Galois Theory?

There are several types of relations in Galois Theory, including equivalence relations, partial orders, and binary operations. Equivalence relations are used to classify objects into distinct classes, while partial orders are used to compare objects based on a certain criteria. Binary operations, on the other hand, are used to combine two elements of a set to produce a third element.

How are relations represented in Galois Theory?

Relations in Galois Theory can be represented in various ways, including tables, graphs, and matrices. Tables are often used to show the mapping between elements of two sets, while graphs are useful for visualizing the connections between objects. Matrices, on the other hand, are used to represent binary operations and their properties.

What is the role of relations in solving algebraic equations?

Relations in Galois Theory are essential for solving algebraic equations. They allow us to determine which operations and transformations can be performed on an equation without changing its solutions. This information is crucial in finding solutions to complicated equations and understanding the symmetries present in their solutions.

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