Exercise · ZW-POLYNOMIAL-RING-005 · ★☆☆ Basic
Which statement about \(\mathbb{Z}[x]\) is true?
Choose one option.
Show answer
- B. is an integral domain
Explanation
The product of two nonzero polynomials with integer coefficients has as leading coefficient the product of the leading ones, which is nonzero; so there are no zero divisors. It is not a field, since \(x\) has no inverse.
Review the concept: Polynomial ring
The polynomial ring \(R[x]\) over a commutative ring \(R\) with identity consists of the finite sums of terms \(a_ix^i\), with coefficients \(a_i\in R\) and an indeterminate \(x\).
\[\Big(\sum_i a_ix^i\Big)\Big(\sum_j b_jx^j\Big)=\sum_k\Big(\sum_{i+j=k}a_ib_j\Big)x^k\]
More practice problems: polynomial ring
- ZW-POLYNOMIAL-RING-001 In ℤ[x], what is (x+1)(x+2)?
- ZW-POLYNOMIAL-RING-002 In ℤ_3[x], what is (x+1)(x+2)?
- ZW-POLYNOMIAL-RING-003 In ℤ[x], what is the coefficient of x^2 in (1+x+x^2)(2+x+3x^2)?
- ZW-POLYNOMIAL-RING-004 In ℤ_2[x], what is (x^2+x+1)+(x+1)?
- ZW-POLYNOMIAL-RING-006 In ℤ[x], f has degree 3 and g has degree 4. What is the degree of fg?
- ZW-POLYNOMIAL-RING-007 In ℤ_6[x], what is the degree of (2x+1)(3x+1)?
- ZW-POLYNOMIAL-RING-008 In ℤ_2[x], what is (x+1)^4?
- ZW-POLYNOMIAL-RING-009 Which of these polynomials is a unit of ℤ_4[x]?