1) Find the factorization of the following polynomials in the given polynomial rings
a) \( x^{4}+4 \) in \( \mathbb{Z}_{5}[x] \)
b) \( x^{3}+2 x^{2}+2 x+1 \) in \( \mathbb{Z}_{7}[x] \)
c) \( x^{2}+x+1 \) in \( \mathbb{Z}_{2}[x] \)
2) Is \( x^{3}+2 x+3 \) an irreducible polynomial of \( \mathbb{Z}_{5}[x] \) ? Express it as a product of irreducible polynomials in \( \mathbb{Z}_{5}[x] \).
3) Is \( 2 x^{3}+x^{2}+2 x+2 \) an irreducible polynomial of \( \mathbb{Z}_{5}[x] \) ? Express it as a product of irreducible polynomials in \( \mathbb{Z}_{5}[x] \).
Answer
3) \( 2x^3+x^2+2x+2 = (x^3+(1/2)x^2+x+1) \), irreducible in \( \mathbb{Z}_{5}[x] \)
Step 1 ：a) \( x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x + 2) \) in \( \mathbb{Z}_{5}[x] \)
Step 2 ：b) \( x^3+ 2x^2 + 2x + 1 = (x+1)(x^2+x+1) \) in \( \mathbb{Z}_{7}[x] \)
Step 3 ：c) \( x^2+x+1 = (x^2+x+1) \) in \( \mathbb{Z}_{2}[x] \)
Step 4 ：2) \( x^3+2x+3 = (x^3+2x+3) \), irreducible in \( \mathbb{Z}_{5}[x] \)
Step 5 ：3) \( 2x^3+x^2+2x+2 = (x^3+(1/2)x^2+x+1) \), irreducible in \( \mathbb{Z}_{5}[x] \)
