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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.
