Do field homomorphisms preserve characteristic

In summary, the statement "Field homomorphisms between fields of different characteristic cannot exist" is false.
  • #1
icantadd
114
0

Homework Statement



Given two fields F,E with different characteristic. Prove or disprove the following statement: "Field homomorphisms between fields of different characteristic cannot exist"

Homework Equations


T : F1 --> F2 is a field homomorphism if
1) T(a+b) = T(a) + T(b)
2) T(ab) = T(a)T(b)
3) T(1) = 1
4) T(0) = 0.


The Attempt at a Solution


Intuition says no...

All field homorphisms are injective. So T:F --> E where F has bigger order than E cannot exist. On the other hand, if E has bigger order than F, F must contain an isomorphic copy of E.

Hmm, not sure where to go from here. Here is my attempt... Suppose we do have a hom from F to E where char E is bigger than char F. Then by the fundamental homomorphism theorem, F/kerT is isomomorphic to E. However since T is injective the kernel is trivial. Therefore F is isomorphic to E contradicting the assumption of different characteristic...
 
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  • #2
icantadd said:

Homework Statement



Given two fields F,E with different characteristic. Prove or disprove the following statement: "Field homomorphisms between fields of different characteristic cannot exist"

Homework Equations


T : F1 --> F2 is a field homomorphism if
1) T(a+b) = T(a) + T(b)
2) T(ab) = T(a)T(b)
3) T(1) = 1
4) T(0) = 0.


The Attempt at a Solution


Intuition says no...

All field homorphisms are injective. So T:F --> E where F has bigger order than E cannot exist. On the other hand, if E has bigger order than F, F must contain an isomorphic copy of E.

Hmm, not sure where to go from here. Here is my attempt... Suppose we do have a hom from F to E where char E is bigger than char F. Then by the fundamental homomorphism theorem, F/kerT is isomomorphic to E. However since T is injective the kernel is trivial. Therefore F is isomorphic to E contradicting the assumption of different characteristic...

That is nonsense. Look up the definition of 'field characteristic'. Read it several times. Then look at your requirement T(1)=1.
 
  • #3
Right read the definitions

[tex]T(1_{F1}) = 1_{F2}[/tex]
Thus,
[tex]T(n1_{F1}) = nT(1_{F1}) = n1_{F2}[/tex]
Thus suppose char(F1) = m,
[tex]T(m1_{F1}) = T(0) = 0 = mT(1_{F1}) = m1_{F2}[/tex]
Therefore, char(F2) <= m.
Suppose p < m satisfies [tex]p1_{F2} = 0[/tex].
Then,
[tex]T^{-1}(p1_{F2}) = T(0) = 0 = T^{-1}(p1_{F2}) = pT^{-1}(1_{F2}) = p1_{F1}[/tex]
Contradicting the minimality of m. Therefore, char(F2) = m.
 
  • #4
P.s. thank you!
 
  • #5
icantadd said:
Right read the definitions

[tex]T(1_{F1}) = 1_{F2}[/tex]
Thus,
[tex]T(n1_{F1}) = nT(1_{F1}) = n1_{F2}[/tex]
Thus suppose char(F1) = m,
[tex]T(m1_{F1}) = T(0) = 0 = mT(1_{F1}) = m1_{F2}[/tex]
Therefore, char(F2) <= m.
Suppose p < m satisfies [tex]p1_{F2} = 0[/tex].
Then,
[tex]T^{-1}(p1_{F2}) = T(0) = 0 = T^{-1}(p1_{F2}) = pT^{-1}(1_{F2}) = p1_{F1}[/tex]
Contradicting the minimality of m. Therefore, char(F2) = m.

Much better. You're welcome!
 

1. What is a field homomorphism?

A field homomorphism is a function that preserves the operations of addition and multiplication between two fields. In other words, if two elements in one field are added or multiplied, the result will be the same as if the operation was performed on the corresponding elements in the other field.

2. How do field homomorphisms relate to characteristic?

Field homomorphisms preserve the characteristic of a field, meaning if the original field has a characteristic of p, then the image field will also have a characteristic of p. In other words, the order of the field is preserved under a homomorphism.

3. Can a field homomorphism preserve characteristic if the fields have different characteristics?

No, a field homomorphism can only preserve the characteristic if the fields have the same characteristic. If the fields have different characteristics, the homomorphism will not preserve the characteristic.

4. How do I know if a field homomorphism preserves characteristic?

You can determine if a field homomorphism preserves characteristic by checking if the order of the fields remains the same after applying the homomorphism. If the order is preserved, then the characteristic is also preserved.

5. Are there any other properties that field homomorphisms preserve?

In addition to preserving the characteristic, field homomorphisms also preserve the identity element, inverse elements, and zero divisors. However, they do not necessarily preserve the multiplicative identity (1) or multiplicative inverses.

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