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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 โ„[๐‘ฅ] be a polynomial ring, then Prove the following
(a) If โ„ is commutative then โ„[๐‘ฅ] is commutative.
(b) If โ„ is a ring with unity then โ„[๐‘ฅ] is a ring with unity.
(c) โ„[๐‘ฅ] is an integral domain if and only if โ„ is an integral domain.
BTL-3 Applying
2. a)
If ๐น is a field and ๐‘“(๐‘ฅ) โˆˆ ๐น[๐‘ฅ] has degree โ‰ฅ 1 , then prove that ๐‘“(๐‘ฅ)
has at most n roots in ๐น. BTL-3 Applying
2. b)
If ๐‘“(๐‘ฅ) = 3๐‘ฅ5 โˆ’ 8๐‘ฅ4 + ๐‘ฅ3 โˆ’ ๐‘ฅ2 + 4๐‘ฅ โˆ’ 7, ๐‘”(๐‘ฅ) = ๐‘ฅ +
9 ๐‘Ž๐‘›๐‘‘ ๐‘“(๐‘ฅ), ๐‘”(๐‘ฅ) โˆˆ โ„ค[๐‘ฅ] , find the remainder when ๐‘“(๐‘ฅ) is divided
by ๐‘”(๐‘ฅ).
BTL-2 Understanding
3
If โ„ is a ring then prove that (โ„[๐‘ฅ], +, . ) is a ring called a
polynomial ring over โ„. BTL-3 Applying
4. a)
Let (โ„ , + , . ) be a commutative ring with unity u. Then โ„
is an integral domain iff for all ๐‘“(๐‘ฅ), ๐‘”(๐‘ฅ) โˆˆ โ„[๐‘ฅ], if neither ๐‘“(๐‘ฅ)
nor ๐‘”(๐‘ฅ) is the zero polynomial, then prove that degree of
๐‘“(๐‘ฅ)๐‘”(๐‘ฅ) = ๐‘‘๐‘’๐‘”๐‘Ÿ๐‘’๐‘’๐‘“(๐‘ฅ) + ๐‘‘๐‘’๐‘”๐‘Ÿ๐‘’๐‘’๐‘”(๐‘ฅ).
BTL-3 Applying
4. b)
Find the remainder when ๐‘”(๐‘ฅ) = 7๐‘ฅ3 โˆ’ 2๐‘ฅ2 + 5๐‘ฅ โˆ’ 2 is divided by
๐‘“(๐‘ฅ) = ๐‘ฅ โˆ’ 3. BTL-2 Understanding
5. a) Find all roots of ๐‘“(๐‘ฅ) = ๐‘ฅ2 + 4๐‘ฅ if ๐‘“(๐‘ฅ) โˆˆ ๐‘12.
5. b)
If ๐‘”(๐‘ฅ) = ๐‘ฅ5 โˆ’ 2๐‘ฅ2 + 5๐‘ฅ โˆ’ 3 & ๐‘“(๐‘ฅ) = ๐‘ฅ4 โˆ’ 5๐‘ฅ3 + 7๐‘ฅ
Find ๐‘ž(๐‘ฅ) , ๐‘Ÿ(๐‘ฅ) ๐‘ ๐‘ข๐‘โ„Ž ๐‘กโ„Ž๐‘Ž๐‘ก ๐‘”(๐‘ฅ) = ๐‘“(๐‘ฅ)๐‘ž(๐‘ฅ) + ๐‘Ÿ(๐‘ฅ).
6. a)
Give an example of polynomial ๐‘“(๐‘ฅ) โˆˆ ๐น(๐‘ฅ), where ๐‘“(๐‘ฅ)
has degree 8 and degree 6, it is reducible but it has no real roots.
6. b) Discuss whether ๐‘ฅ4 โˆ’ 2 is reducible over โ„š , โ„ , โ„‚.
7. a) State and Prove Factor Theorem. BTL-3 Applying
7. b)
Determine whether the given polynomial is irreducible or not?
๐‘“(๐‘ฅ) = ๐‘ฅ2 + ๐‘ฅ + 1 over ๐‘3, ๐‘5, ๐‘7
8. Show that: A finite field F has order ๐‘๐‘ก where p is a prime ๐‘ก โˆˆ ๐‘ง+.

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

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