Can GPT4 help you learn anything?

I have a theory that with the correct explanation, you can learn anything.?

I’ve been self-testing this theory using GPT4 and a math concept I don’t understand and don’t have the prerequisites to understand: Galois Theory.????

While I’m continuing to experiment with prompts, the response you may want to try is the last one: can you generate a metaphor for automorphism, and rewrite the above example using that metaphor

GPT4’s response was incredible.? It not only generated a novel, but commonly understandable metaphor, it also rewrote its previous, expertise-requiring answer integrating that metaphor.??

While I wouldn’t be able to pass a university class with the limited understanding the responses provide, it does show, in my opinion, how the correct prompts can radically accelerate understanding.?

Additionally, now that GPT4 has allowed for plugins, the Show Me plugin gives the ability to generate concept, attribute, and behavior maps.

#chatgpt #gpt4 #promptengineering #math #ai #learning #education

Sean

what are the key concepts of galois theory

GPT4

Galois Theory is a branch of abstract algebra that studies the relationship between fields and the polynomial equations they solve. It was developed by évariste Galois, a French mathematician, in the 19th century. Here are some of the key concepts in Galois Theory:

1. **Fields:** In algebra, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is a fundamental algebraic structure, and Galois Theory studies fields and their extensions.

2. **Field Extensions:** A field extension is a pair of fields such that the operations of the smaller field are those of the larger field restricted to the smaller field. In Galois Theory, field extensions are used to solve polynomial equations.

3. **Polynomials:** A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Galois Theory provides a connection between the roots of a polynomial and the structure of a field extension generated by those roots.

4. **Galois Group:** The Galois group of a polynomial is a certain group associated with the polynomial. The group's structure reveals key information about the roots of the polynomial and their relationships.

5. **Automorphisms:** An automorphism is a bijective map from a mathematical object to itself that preserves the structure of the object. Field automorphisms play a crucial role in Galois Theory.

6. **Solvability by Radicals:** One of the most important results of Galois Theory is a criterion that determines whether a polynomial is solvable by radicals (i.e., whether the roots of the polynomial can be expressed using only the operations of addition, subtraction, multiplication, division, and taking roots). This was famously used to prove that there is no general solution in radicals to polynomial equations of degree five or higher.

These concepts are interconnected, and Galois Theory provides a framework for understanding their relationships and using them to solve problems in algebra and number theory.

Sean

how is automorphism used?

GPT4

Automorphisms are used extensively in various fields of mathematics to study the structure of mathematical objects. Here are a few examples:

1. **Algebra:** In algebra, automorphisms are used to study the structure of algebraic structures such as groups, rings, and fields. For example, the set of all automorphisms of a group forms a group itself, known as the automorphism group of the group. This can provide insight into the structure and properties of the original group.

2. **Graph Theory:** In graph theory, an automorphism of a graph is a permutation of the vertices that preserves adjacency. Studying graph automorphisms can provide information about the symmetry and structure of the graph.

3. **Differential Geometry:** In differential geometry, automorphisms are used to study the structure of differentiable manifolds. For example, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.

4. **Number Theory:** In number theory, the Galois group of a polynomial is an example of an automorphism group. It consists of field automorphisms of the splitting field of the polynomial, and its structure provides information about the roots of the polynomial.

5. **Topology:** In topology, homeomorphisms are a type of automorphism. A homeomorphism is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space.

In all these cases, automorphisms are used to reveal the underlying structure and symmetries of mathematical objects. They are a fundamental tool in many areas of mathematics.

Sean

can you give me an example of automorphism used in number theory?

GPT4

Sure, a classic example of automorphisms in number theory is found in the study of Galois groups.

Consider a polynomial equation with coefficients in a field (for example, the field of rational numbers). If the polynomial has degree greater than 4, there is no general formula to express its roots in terms of radicals. However, we can still gain a lot of information about the roots by considering the field extension generated by the roots of the polynomial.

The Galois group of the polynomial is defined as the group of field automorphisms of this field extension that fix the original field. Each automorphism in the Galois group permutes the roots of the polynomial.

The structure of the Galois group can tell us a lot about the roots of the polynomial. For example, if the Galois group is abelian (i.e., all its elements commute), then the roots of the polynomial can be expressed in terms of radicals. This is the Fundamental Theorem of Galois Theory.

Here's a concrete example. Consider the polynomial equation x^2 - 2 = 0. The roots of this polynomial are sqrt(2) and -sqrt(2). The field extension generated by these roots is the field of all numbers of the form a + b*sqrt(2), where a and b are rational numbers.

The Galois group of this polynomial is a group of two elements: the identity automorphism (which sends each element to itself), and the automorphism which sends a + b*sqrt(2) to a - b*sqrt(2). This automorphism permutes the roots of the polynomial, sending sqrt(2) to -sqrt(2) and vice versa.

This Galois group is abelian (in fact, it's isomorphic to the group of integers modulo 2), so the roots of the polynomial can be expressed in terms of radicals, which we already knew since the polynomial is quadratic.

Then it gets interesting...

Sean

can you generate a metaphor for automorphism, and rewrite the above example using that metaphor

GPT4

Sure, let's use the metaphor of a puzzle. Imagine you have a puzzle where each piece has a unique shape and fits in a specific place. Now, imagine you have a magical power that allows you to rearrange the pieces of the puzzle while still maintaining the overall picture. This magical power is like an automorphism.

An automorphism is a transformation that rearranges the elements of a structure while preserving the overall structure. Just like how you can rearrange the puzzle pieces but still keep the same picture, an automorphism rearranges the elements of a mathematical object (like a group, field, or graph) while preserving its structure.

Now, let's revisit the number theory example using this metaphor.?

Imagine you have a puzzle that represents a polynomial equation, say x^2 - 2 = 0. The pieces of this puzzle are the roots of the equation, which are sqrt(2) and -sqrt(2).?

Now, you use your magical power (the automorphism) to rearrange the puzzle pieces. You swap the piece representing sqrt(2) with the piece representing -sqrt(2). Even though the pieces have moved, the overall picture of the puzzle (the solution to the equation) remains the same.?

This is what an automorphism does in the context of a Galois group in number theory. It rearranges (permutes) the roots of the polynomial, but it does so in a way that preserves the overall structure (the solution to the equation).?

And just like how the way you can rearrange the puzzle pieces can tell you something about the puzzle itself, the structure of the Galois group (the set of all possible automorphisms) can tell you something about the solutions to the polynomial equation.