# Is the following a field?

I think I may need a refresher in logs here. The question is:

F=$\{a \in R \vert a<1\} 1<t \in R$

(1)$a\#b= a+b-ab$ for all $a,b \in$ F

(2)$a*b=1-t^{log_t(1-a) * log_t(1-b)}$ for all $a,b \in F$ where # and * are just function notation.

Is this a field?

Well, starting with (1), I stated that a=1/2 and b=2/3, then

$1/2\#2/3= 1/2 + 2/3 - (1/2)(2/3)= 5/6$ which is less than 1

(2) then is $(1/2)*(2/3)=1-t^{log_t(1-a) log_t(1-b)}$

Here is where I'm lost on how to work the rest. I'm pretty sure its going to turn out to be less than 1 since we have 1-something bigger than 1. After, this I believe I then have to use the 6 conditions that determine a field. They are:

1. associativity of addition and multiplication
2. commutativity of addition and multiplication
3. distributivity of multiplication over addition
4. existence of identity elements for addition and multiplication
• Try finding a map that converts $\#$ and $\ast$ into ordinary addition and multiplication. – Daniel Fischer Jun 5 '14 at 20:38
• another hint along Daniel Fischer's lines: $a\# b = 1-(1-a)(1-b)$. – Dustan Levenstein Jun 9 '14 at 20:32
• It seems to me condition, or rather definition, $(2)$ is nested. The operation $*$ appears on the left hand side as well as in the exponent in the right hand side. Continued exponential(similar to continued fractions)? – ReverseFlow Jun 14 '14 at 4:26
• @Genomeme According to answers below it seems in $(2)$ it is the standard multiplication in the exponent – Arun Kumar Jun 16 '14 at 16:18
Consider the bijective mapping $$\psi:F\to\mathbb{R},\psi(x)=\log_t(1-x)$$ with $\psi^{-1}(x)=1-t^x$. Then \eqalign{ a\#b&=\psi^{-1}(\psi(a)+\psi(b))\cr a*b&=\psi^{-1}(\psi(a)\cdot\psi(b))} This proves that $\psi$ transfers the field structure of $(\mathbb{R},+,0)$ to $(F,\#,*)$. So, $(F,\#,*)$ is a field isomorphic to the field of real numbers.
• This is a general principle called, transfer of structure, you needn't verify all the properties, but it is a good exercise to see using the definition how properties of $\mathbb{R}$ are transferred to $F$. – Omran Kouba Jun 11 '14 at 16:28