Polynomials in Z6[x]: Find & Explain Deg 0 Product

[sarah77]

Homework Statement

Find two polynomials, each of degree 2, in Z6[x] whose product has degree 0. Can you repeat the same in Z7[x]? Explain.

Homework Equations

In Z6[x] and Z7[x] can the only variable be x?

The Attempt at a Solution

I know Z6 consists of {0,1,2,3,4,5} and Z7: {0,1,2,3,4,5,6}; and I have tried (x2+5)(x2-3) and others but I get degree of 4, and it must be degree of 6 for it to be degree 0 in Z6..Please help, am I confused on how to solve this.

 

[Dick]

What's 3x^2 times 2x^2 in Z6?

 

[sarah77]

6x^4, so 0x^4...the polynomial is of degree 4, but since the coefficient is 0, the product would have a degree of 0?

 

[Dick]

6x^4, so 0x^4...the polynomial is of degree 4, but since the coefficient is 0, the product would have a degree of 0?

Sure. Now why can't that happen in Z7?

 

[sarah77]

Since it is a prime number, no two elements in Z7 can be multiplied to obtain a number divisible by 7.

 

[Dick]

Since it is a prime number, no two elements in Z7 can be multiplied to obtain a number divisible by 7.

Exactly. There are no zero divisors in Z7. There are in Z6.

 

[sarah77]

Thank you, that makes sense!