- #1

ti89fr33k

- 13

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I've tried to see it as a factor group, but I'm stuck. Can someone help?

mary

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In summary, the kernel of a field homomorphism is the set of all elements in the domain that map to the identity element in the codomain. It is an important component of a homomorphism, determining which elements in the domain are mapped to the identity element. In abstract algebra, the kernel allows for the classification and study of different types of homomorphisms. It can be empty, typically in the case of an isomorphism, and is also relevant in the study of field extensions. It helps to determine if a field extension is simple or complex.

- #1

ti89fr33k

- 13

- 0

I've tried to see it as a factor group, but I'm stuck. Can someone help?

mary

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- #2

matt grime

Science Advisor

Homework Helper

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I'm sorry, are you think ing of kernels as morphisms or objects? Cos you refer to the kernel as a homomorphism and some thing that is isomorphic to the field.

In any case the kernel of a ring homomorphism (of which a field homomrphism is a special case) is an ideal. So how many ideals of a field are there?

In any case the kernel of a ring homomorphism (of which a field homomrphism is a special case) is an ideal. So how many ideals of a field are there?

Last edited:

- #3

M98Ranger

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The kernel of a field homomorphism is an important concept in abstract algebra. It is defined as the set of all elements in the domain that are mapped to the identity element in the codomain. In other words, it is the set of elements that are mapped to zero under the homomorphism.

To show that the kernel of a field homomorphism is either the trivial homomorphism or isomorphic to the field, we need to consider two cases:

1. The trivial homomorphism: In this case, the kernel is simply the set containing only the identity element of the domain. This is because the only element that is mapped to the identity element in the codomain is the identity element itself. Therefore, the kernel is isomorphic to the trivial homomorphism.

2. Non-trivial homomorphism: In this case, the kernel is a proper subgroup of the domain. This is because the homomorphism preserves the field operations, so the identity element of the codomain must also be the identity element of the domain. Therefore, the kernel cannot be the entire domain.

Now, since the kernel is a proper subgroup of the domain, it must have a non-zero element. Let's call this element "a." Since a is in the kernel, it must be mapped to the identity element in the codomain, which we will call "0."

Now, consider the set of all multiples of a, denoted by <a>. Since the field homomorphism preserves multiplication, the set <a> is also mapped to the identity element 0 in the codomain. This means that <a> is a proper ideal of the field, and therefore, it must be the entire field.

Hence, the kernel of a non-trivial field homomorphism is isomorphic to the field itself. This completes the proof that the kernel of a field homomorphism is either the trivial homomorphism or isomorphic to the field.

The kernel of a field homomorphism is the set of all elements in the domain that map to the identity element in the codomain.

The kernel is an important component of a homomorphism as it determines which elements in the domain are mapped to the identity element in the codomain.

The kernel is significant in abstract algebra as it allows for the classification and study of different types of homomorphisms, including ring homomorphisms and group homomorphisms.

Yes, it is possible for the kernel of a field homomorphism to be empty. This typically occurs when the homomorphism is an isomorphism, where all elements in the domain are mapped to unique elements in the codomain.

The kernel of a field homomorphism is an important concept in the study of field extensions. It helps to determine if a field extension is a simple extension, where the kernel is trivial, or a more complex extension, where the kernel is not trivial.

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