**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[𝑥] be a polynomial ring, then Prove the following

(a) If ℝ is commutative then ℝ[𝑥] is commutative.

(b) If ℝ is a ring with unity then ℝ[𝑥] is a ring with unity.

(c) ℝ[𝑥] is an integral domain if and only if ℝ is an integral domain.

BTL-3 Applying

2. a)

If 𝐹 is a field and 𝑓(𝑥) ∈ 𝐹[𝑥] has degree ≥ 1 , then prove that 𝑓(𝑥)

has at most n roots in 𝐹. BTL-3 Applying

2. b)

If 𝑓(𝑥) = 3𝑥5 − 8𝑥4 + 𝑥3 − 𝑥2 + 4𝑥 − 7, 𝑔(𝑥) = 𝑥 +

9 𝑎𝑛𝑑 𝑓(𝑥), 𝑔(𝑥) ∈ ℤ[𝑥] , find the remainder when 𝑓(𝑥) is divided

by 𝑔(𝑥).

BTL-2 Understanding

3

If ℝ is a ring then prove that (ℝ[𝑥], +, . ) 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 𝑓(𝑥), 𝑔(𝑥) ∈ ℝ[𝑥], if neither 𝑓(𝑥)

nor 𝑔(𝑥) is the zero polynomial, then prove that degree of

𝑓(𝑥)𝑔(𝑥) = 𝑑𝑒𝑔𝑟𝑒𝑒𝑓(𝑥) + 𝑑𝑒𝑔𝑟𝑒𝑒𝑔(𝑥).

BTL-3 Applying

4. b)

Find the remainder when 𝑔(𝑥) = 7𝑥3 − 2𝑥2 + 5𝑥 − 2 is divided by

𝑓(𝑥) = 𝑥 − 3. BTL-2 Understanding

5. a) Find all roots of 𝑓(𝑥) = 𝑥2 + 4𝑥 if 𝑓(𝑥) ∈ 𝑍12.

5. b)

If 𝑔(𝑥) = 𝑥5 − 2𝑥2 + 5𝑥 − 3 & 𝑓(𝑥) = 𝑥4 − 5𝑥3 + 7𝑥

Find 𝑞(𝑥) , 𝑟(𝑥) 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑔(𝑥) = 𝑓(𝑥)𝑞(𝑥) + 𝑟(𝑥).

6. a)

Give an example of polynomial 𝑓(𝑥) ∈ 𝐹(𝑥), where 𝑓(𝑥)

has degree 8 and degree 6, it is reducible but it has no real roots.

6. b) Discuss whether 𝑥4 − 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?

𝑓(𝑥) = 𝑥2 + 𝑥 + 1 over 𝑍3, 𝑍5, 𝑍7

8. Show that: A finite field F has order 𝑝𝑡 where p is a prime 𝑡 ∈ 𝑧+.

Subject name | ALGEBRA AND NUMBER THEORY |

Short Name | ANT |

Semester | 5 |

Subject Code | MA8551 |

Regulation | 2017 regulation |

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