Using the First Isomorphism Theorem

In summary, the first isomorphism theorem for groups states that two groups are isomorphic if there exists a homomorphism between them with a specific kernel. An example of this is shown with Q[x]/(x^2-3) and {a+b*sqrt(3)}, where the homomorphism sends x to sqrt(3). To find the homomorphism, one needs to find an element whose square is equal to the kernel, and then use that element to construct the isomorphism.
  • #1
CurtBuck
5
0
I'm trying to understand the first isomorphism theorem for groups.

Part of the examples given in the book is showing that Q[x]/(x^3-3) is isomorphic to {a+b*sqrt(3)}

As I understand it, by finding a homomorphism from Q[x] to {a+b*sqrt(3)} in which the kernel is x^3-3, the two are isomorphic.

I am struggling in finding the homomorphism from Q[x] to {a+b*sqrt(3)}

Any help would be great.
 
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  • #2
Are you sure it's not supposed to be Q[x]/(x2-3)?
 
  • #3
Yeah, it was supposed to be Q[x]/(x^2-3)
 
  • #4
You need to find an element in Q[x]/(x2-3) whose square is equal to 3, i.e. you want to find some y such that y2+(x2-3) = 3+(x2-3) (y is a polynomial here). So there's really two steps here

1) Find y.
2) Use y to construct an isomorphism
 
  • #5
the homomorphism sends x to sqrt(3).

when you mod out by f(x), you make f(x) = 0, so x becomes a root of f. since sqrt(3) is the root of x^2-3, x becomes sqrt(3).
 

What is the First Isomorphism Theorem?

The First Isomorphism Theorem is a fundamental result in group theory that states that if f is a group homomorphism from one group G to another group H, then the kernel of f is a normal subgroup of G, and the image of f is isomorphic to the quotient group G/ker(f).

How is the First Isomorphism Theorem used in mathematics?

The First Isomorphism Theorem is used to simplify group structures and make them easier to study. It is also used to establish connections between different groups and to prove other theorems in group theory.

What is the significance of the First Isomorphism Theorem?

The First Isomorphism Theorem is significant because it allows us to better understand and analyze groups by breaking them down into simpler, isomorphic structures. It also provides a powerful tool for proving other theorems in group theory.

Can the First Isomorphism Theorem be applied to other mathematical structures?

The First Isomorphism Theorem can be applied to other algebraic structures, such as rings and modules, as long as the appropriate definitions for homomorphisms, kernels, and images are used.

What are some real-world applications of the First Isomorphism Theorem?

The First Isomorphism Theorem has applications in many areas of mathematics and science, including cryptography, coding theory, and physics. It is also used in computer science for data compression and error correction algorithms.

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