Factoring x^5+x+1 Using Modulo - A Guide

[amcavoy]

How can I factor $$x^5+x+1$$ using modulo? I know, for example, I could write $$(x+3)(x+5)$$ as $$x^2+x+1 mod7$$. How can I go backwards with this?

Thanks.

 

[matt grime]

You can't.

 

[tacman]

To factor x^5+x+1 using modulo, you can follow these steps:

1. First, we need to find a prime number p that is relatively prime to the coefficients of x^5+x+1. In this case, we can choose p=7.

2. Next, we can rewrite x^5+x+1 as (x^5+6x^3+5x^2+6x+1) mod7. This is because we can replace any term with its equivalent modulo p, and 1 mod7 is just 1.

3. Now, we can use the factoring method of difference of squares. We can rewrite (x^5+6x^3+5x^2+6x+1) as ((x^2)^2x+x^2+1) mod7.

4. This can be further simplified as ((x^2)^2x+(x^2+1)) mod7.

5. Now, we can factor out x^2 from the first term to get (x^2(x^2+1)) mod7.

6. Finally, we can factor (x^2+1) into (x+3)(x+5) mod7. This is the same as (x+3)(x+5) in the original equation.

Therefore, the factored form of x^5+x+1 using modulo 7 is (x+3)(x+5).