**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 |

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