Proving Isomorphism of Fields with Four Elements

• Bashyboy
In summary, the conversation discusses how to prove that any two fields with four elements are isomorphic. The conversation explores two possible methods of proof, one involving expanding (1 + 1)^2 and the other involving an assumption about a and b not being equal to 0. Both methods lead to the conclusion that a + b must equal 0, or b = -a, and therefore (-a)^3 = 1, which implies -1 = 1. This contradiction proves the claim that any two fields with four elements are isomorphic.
Bashyboy

Homework Statement

Show that any two Fields with four elements are isomorphic.

The Attempt at a Solution

Let ##F= \{0,1,a,b\}## be a field with four elements. Since ##F## is an additive abelian group, by Lagrange's theorem ##1## must have either an order of ##2## or ##4##; i.e., either ##1+1=0## or ##1+1+1+1=0##. Since ##F## is a field, every nonzero element is a unit, meaning that the group of units ##U(F)## is a group of ##3## which makes it isomorphic to ##\Bbb{Z}_3##. Note that neither ##a+b=a## nor ##a+b=b## can hold. This leaves us with ##a+b = 0## or ##a+b=1##...This is where I get stuck. I am trying to show that ##1+1 = 0##. If I can rule out ##a+b=1##, then I can show ##b=-a##, and therefore ##(-a)^3 = 1## or ##-1=1## or ##1+1=0##. Another route is to assume that ##1+1+1+1=0## is true, which would imply ##F \simeq \Bbb{Z}_4##, and then deduce a contradiction, but I haven't been able to identify the contradiction. Perhaps there is a problem with ##F \simeq \Bbb{Z}_4## and ##U(F) \simeq \Bbb{Z}_3## being simultaneously true...I could use some guidance.

What do you get if you expand $(1 + 1)^2$?

Okay. So ##(1+1)^2 = (1+1)(1+1) = 1 + 1 + 1 + 1 = 0##. Since we are working in a field there can be no nonzero divisors which means ##1+1=0##.

Would this be another valid way of proving the claim? Suppose that ##a+b \neq 0##. This means that ##a+b \in U(F) \simeq \Bbb{Z}_3##, which implies that ##(a+b)^3 = 1## or ##1 + 3a^2b + 3a b^2 = 0##. Multiplying by ##a## and then ##b## yields the equations ##a + 3b + 3a^2 b^2 = 0## and ##b + 3a^2 b^2 + 3a = 0##, and adding the two gives ##4a + 4b + 6a^2 b^2 = 0## or ##6a^2 b^2 = 0##, which is a contradiction since neither ##a=0## nor ##b=0##. Hence ##a + b = 0## or ##b = -a##. Then ##(-a)^3 = 1## implies ##-1 = 1##.

Admittedly it is a bit longer, but somewhat interesting, although it may be invalid/unsound.

Bashyboy said:
Okay. So ##(1+1)^2 = (1+1)(1+1) = 1 + 1 + 1 + 1 = 0##. Since we are working in a field there can be no nonzero divisors which means ##1+1=0##.

Would this be another valid way of proving the claim? Suppose that ##a+b \neq 0##. This means that ##a+b \in U(F) \simeq \Bbb{Z}_3##, which implies that ##(a+b)^3 = 1## or ##1 + 3a^2b + 3a b^2 = 0##. Multiplying by ##a## and then ##b## yields the equations ##a + 3b + 3a^2 b^2 = 0## and ##b + 3a^2 b^2 + 3a = 0##, and adding the two gives ##4a + 4b + 6a^2 b^2 = 0## or ##6a^2 b^2 = 0##, which is a contradiction since neither ##a=0## nor ##b=0##.

How do you justify the implicit assumption that $6 \neq 0$?

Hence ##a + b = 0## or ##b = -a##. Then ##(-a)^3 = 1## implies ##-1 = 1##.

It is in fact true that $a + b = 1$: it follows from $1 + 1 = 0$ and the field axioms.

1. What are the four elements in the "Field with Four Elements"?

The four elements in the "Field with Four Elements" are earth, water, air, and fire.

2. How do these four elements interact with each other in the field?

The four elements interact with each other in a cyclical manner. Earth creates water, water nourishes plants, plants produce air, and air fuels fire. Fire then creates earth by burning plants and vegetation.

3. What is the significance of the four elements in the field?

The four elements represent the natural forces that sustain life on Earth. They symbolize balance, harmony, and interconnectedness in nature.

4. Can these four elements be found in other fields or environments?

Yes, the four elements can be found in various forms in different environments. For example, earth can be seen as soil or rocks, water can be found in oceans or rivers, air can be felt as wind, and fire can manifest as lightning or sunlight.

5. How can we apply the concept of the four elements in scientific research?

The concept of the four elements can be used as a framework for understanding and studying natural phenomena. It can also be applied in fields such as chemistry, geology, and meteorology to explain the properties and interactions of different substances and materials.

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