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# MA8551 Question Bank ALGEBRA AND NUMBER THEORY Regulation 2017 Anna University

## MA8551 Question Bank ALGEBRA AND NUMBER THEORY

MA8551 Question Bank ALGEBRA AND NUMBER THEORY Regulation 2017 Anna University free download. ALGEBRA AND NUMBER THEORY Question Bank MA8551 pdf free download.

### Sample MA8551 Question Bank ALGEBRA AND NUMBER THEORY

Let ℝ[????] be a polynomial ring, then Prove the following
(a) If ℝ is commutative then ℝ[????] is commutative.
(b) If ℝ is a ring with unity then ℝ[????] is a ring with unity.
(c) ℝ[????] is an integral domain if and only if ℝ is an integral domain.
BTL-3 Applying
2. a)
If ???? is a field and ????(????) ∈ ????[????] has degree ≥ 1 , then prove that ????(????)
has at most n roots in ????. BTL-3 Applying
2. b)
If ????(????) = 3????5 − 8????4 + ????3 − ????2 + 4???? − 7, ????(????) = ???? +
9 ???????????? ????(????), ????(????) ∈ ℤ[????] , find the remainder when ????(????) is divided
by ????(????).
BTL-2 Understanding
3
If ℝ is a ring then prove that (ℝ[????], +, . ) is a ring called a
polynomial ring over ℝ. BTL-3 Applying
4. a)
Let (ℝ , + , . ) be a commutative ring with unity u. Then ℝ
is an integral domain iff for all ????(????), ????(????) ∈ ℝ[????], if neither ????(????)
nor ????(????) is the zero polynomial, then prove that degree of
????(????)????(????) = ????????????????????????????(????) + ????????????????????????????(????).
BTL-3 Applying
4. b)
Find the remainder when ????(????) = 7????3 − 2????2 + 5???? − 2 is divided by
????(????) = ???? − 3. BTL-2 Understanding
5. a) Find all roots of ????(????) = ????2 + 4???? if ????(????) ∈ ????12.
5. b)
If ????(????) = ????5 − 2????2 + 5???? − 3 & ????(????) = ????4 − 5????3 + 7????
Find ????(????) , ????(????) ????????????ℎ ????ℎ???????? ????(????) = ????(????)????(????) + ????(????).
6. a)
Give an example of polynomial ????(????) ∈ ????(????), where ????(????)
has degree 8 and degree 6, it is reducible but it has no real roots.
6. b) Discuss whether ????4 − 2 is reducible over ℚ , ℝ , ℂ.
7. a) State and Prove Factor Theorem. BTL-3 Applying
7. b)
Determine whether the given polynomial is irreducible or not?
????(????) = ????2 + ???? + 1 over ????3, ????5, ????7
8. Show that: A finite field F has order ???????? where p is a prime ???? ∈ ????+.

 Subject name ALGEBRA AND NUMBER THEORY Short Name ANT Semester 5 Subject Code MA8551 Regulation 2017 regulation