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## Re: [Axiom-developer] Zero divisors in Expression Integer

 From: William Sit Subject: Re: [Axiom-developer] Zero divisors in Expression Integer Date: Thu, 04 Jan 2007 01:31:13 -0500

```On Thu, 4 Jan 2007 05:47:29 +0100 (CET)
```
I have already written that due to incomplte simplification we may get zero divisors in Expression Integer. Below an easy example that multiplication in Expression Integer is nonassociative
```(or, if you prefer, a proof that 1 equals 0):

(135) -> c1 := sqrt(2)*sqrt(3*x)+sqrt(6*x)

+--+    +-+ +--+
(135)  \|6x  + \|2 \|3x
```
Type: Expression Integer
```(136) -> c2 := sqrt(2)*sqrt(3*x)-sqrt(6*x)

+--+    +-+ +--+
(136)  - \|6x  + \|2 \|3x
```
Type: Expression Integer
```(137) -> (1/c1)*c1*c2*(1/c2)

(137)  1
```
Type: Expression Integer
```(138) -> (1/c1)*(c1*c2)*(1/c2)

(138)  0
```
Type: Expression Integer
```
```
But this is not just an Axiom problem. Mathematica does the same thing, with a slight variation on input: a1 = Sqrt*Sqrt[3 Sqrt[5x + 7] + 6] - Sqrt[6Sqrt[5x + 7] + 12] a2 = Sqrt*Sqrt[3 Sqrt[5x + 7] + 6] + Sqrt[6Sqrt[5x + 7] + 12]
```(1/a1)*a1*a2*(1/a2)  (* answer 1 *)
```
(1/a1)*(a1*a2 // Simplify)*(1/a2) (*answer 0, Simplify is needed to get this *)
```
```
The problem seems to be the lack of a canonical form for radical expressions and an algorithm to reduce expressions to canonical form. A related problem is lack of algorithm to test zero. Another is denesting of a nested radical expression. These problems have been studied by Zippel, Landau, Tulone et al, Carette and others.
```
William

```