Question

In: Advanced Math

6. Find all monic irreducible polynomials of degree ≤ 3 over Z3. Using your list, write...

6. Find all monic irreducible polynomials of degree ≤ 3 over Z3. Using your list, write each of the following polynomials as a product of irreducible polynomials over Z3:

(a) x^4 + 2x^2 + 2x + 2.

(b) 2x^3 − 2x + 1.

(c) x^4 + 1.

Expert Solution

We are to enumerate all the irreducible monic polynomials with degree over the field . First observe that if any polynomial of degree less than is reducible, it must have a degree one factor, thus in particular the polynomial must have a root over the field. Hence an easy way to check for irreducibility is to check whether we have a root or not.

Degree 1 Polynomials

These are easy to write down. We have 3 possibilities. Namely,

Degree 2 Polynomials

Any degree 2 monic polynomial looks like . Since it cannot have as a root, we must have that . Thus we need to put and check whether we have any roots. We have 6 polynomials to check for. An easy calculation gives the following list of irriducible polynomials.

Degree 3 Polynomials

The strategy is same as before. We have any monic degree 3 polynomial looks like . Since 0 cannot be a root, we must have . Thus we need to put the values and check for root. There are 18 polynomials to check for! Another easy (albeit lengthy) calculation gives the following list of irreducible polynomials.

Lastly we need to factorize the given polynomials.

Part (a)

Given . We check, , since we are working over . Thus 2 is a root and we find out, . But we again see that 2 is a root of the second factor and hence we further factorize, . These are the irreducible factors.

Part (b)

Given, . We see that this polynomial has no roots over the field. Thus it must be a scalar multiple of some degree 3 irreducible polynomial. Since over we see, , which is the irreducible factorization.

Part(c)

Given, . Again we have that this polynomial has no roots over the field. Then either it is irreducible itself or product of two degree 2 irreducible polynomials. Since there are only 3 such polynomials, as listed above, we can simply check their products. We can then see that, , which is the irreducible factorization.

Colby Messinger answered 2 weeks ago

how many irreducible polynomials of degree 2 in Z3 [x]

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