Discussion
Which one of the following statements is TRUE?
Answer: C
The true statement is that the polynomials with real coefficients form a Unique Factorization Domain.
\(\mathbb{R}[x]\) is a polynomial ring over a field, hence a Euclidean domain (division with remainder by degree), hence a principal ideal domain and so a UFD. Every nonzero non-unit factors uniquely into irreducibles, which over \(\mathbb{R}\) are linear and irreducible quadratic factors.
The other rings are not even integral domains. \(2 \times 2\) real matrices are not commutative and have zero divisors. Real valued functions on a set with two or more points have zero divisors: a function vanishing on one half times one vanishing on the other is 0. The same construction works for Riemann integrable functions on [0,1].
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development