Abstract Algebra II · Polynomial rings and irreducibility
Content and primitive part
Problem
For $f(x)=12x^3-18x^2+30\in\mathbb Z[x]$, find a positive content $c(f)$ and a primitive polynomial $f_0$ with $f=c(f)f_0$. Explain how the answer changes if content is allowed to be negative.
Hint
Compute $\gcd(12,18,0,30)$.
Check your work
Work the problem yourself first. Then open it in Training to check your answer and read the full worked solution.
The answer check and full solution for this problem come with ProofAnvil Practice membership ($19 USD monthly). See membership. Or start with the free Abstract Algebra II sample problem: Try the free sample problem.
More Abstract Algebra II practice problems
- Gaussian integers form a subringRings, ideals, quotient rings, and homomorphisms
- Kernel of evaluation at a pointRings, ideals, quotient rings, and homomorphisms
- Divisibility is transitiveDomains, divisibility, and factorization
- Lcm from prime exponentsDomains, divisibility, and factorization
- Reciprocal polynomials preserve irreducibilityPolynomial rings and irreducibility
- Kernels and images are submodulesModules and finitely generated modules over a PID
- When finitely generated torsion means finiteModules and finitely generated modules over a PID
- A cubic extension has no quadratic intermediate fieldAlgebraic field extensions and minimal polynomials
- A quartic whose splitting field is only quadraticFinite Galois theory