Metamath

Hot, Cold And Lukewarm
The Klein four group has a cyclic multiplicative group in GF(4). The group GF(4)* is isomorphic to C3 and the field has its prime subfield of [0,1] in common with the field GF(8). That results in the intersection of the two and the remaining two elements of GF(4) being "cut off".
C3 we used before in the "2, 3, 4 cycle" from the previous page, although we used addition on {0,1,2} rather than multiplication on {1, x, x^{2}}, as the set used  however both are isomorphic.
GF(4) is generally written as [0, 1, x, x+1] where addition is done modulo 2 and the indeterminate "x" is under the irreducible formula of x^{2} + x + 1 = 0. We may state this as if "1" were a vector, "x" were a second vector and that "x+1" was a vector in the direction of the sum of "1 and x".
We then would state that "1" is "hot" , "x" is cold, and the sum or mingling of the two "x + 1" is "lukewarm".
Of course when we extrapolate to the octal we have in the K4 form of the ultrafilter,
a = 1 = [0,a,b,c]
b = [0,a,d,e]
c = [0,a,f,g]
And the "two parts" cut off are congruent to b and c, which in the octal is {d,e,f,g}.
Of course there is the frobenius map whcih sends GF(4) to an automorphism. It becomes plain that to be cold is to be equivalent to lukewarm, wher the map sends x => x^{2} and x^{2} => x.
In the octal ultrafitler though we could have;
e = 1 = [0,a,b,c]
g = [0,a,d,e]
c = [0,a,f,g]
There is no sense that untity may contain itself in the octal, and the "two parts cut off" may only be equated to {d,e,f,g} if [a,b,c] is held static under frobenius and one of {d,e,f,g} must be unity. In the octal example above we have e = 1 and [b,d,f] held static.
The result is that only the intersection in the field GF(8) (when a = 1) with GF(4) is structurally sound in both. [0,1] = [0,a] is this intersection, and the remaining elements of [a,b,c] are cut off from the octal, or equivalently we see that the static subgroup of GF(8) must not be a group found in Christ's K4 form, un;less we multiply under C7 to a different choice of unity.
Either way, two parts are "cut off". Scripture shows that these parts are the "cold and lukewarm".
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