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Chapter 4: Q.4.4.3-23E (page 104)

Question:
Show that $x^{2} + 2$ is irreducible in $Z_{5} [x]$ .
Factor $x^{4} - 4$ as a product of irreducible in $Z_{5} [x]$ .

Short Answer

Expert verified

Answer:

It is proved that $x^{2} + 2$ is irreducible in $Z_{5} [x]$ .
The required factors $x^{3} + 3$ and $x^{2} + 2$ are irreducible in $Z_{5} [x]$ .

Step by step solution

Property of Polynomial:

Polynomial Operations:

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Most popular questions from this chapter

Prove Corollary 4.9.

Let R be a commutative ring with identity and a $\in$ R.

If $a^{3} = 0_{R}$ . Show that $1_{R} + a x$ is a unit in R[x].

If $a^{4} = 0_{R}$ , Show that $1_{R} + a x$ is a unit in R[x].

Use unique factorization to find the gcd in $ℂ [x]$ of ${(x - 3)}^{3} {(x - 4)}^{4} {(x - i)}^{2}$ and $(x - 1) (x - 3) {(x - 4)}^{3}$ .

If $c \in R$ is a zero divisor in a commutative ring R , then c is also a zero divisor in R[x].

Show that $\sqrt{p}$ is irrational for every positive prime integer p. [Hint: What are the roots of $x^{2} - p^{2}$ ? Do you prefer this proof to the one in Exercises 30 and 31 of Section 1.37].

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Question:
Show that $x^{2} + 2$ is irreducible in $Z_{5} [x]$ .
Factor $x^{4} - 4$ as a product of irreducible in $Z_{5} [x]$ .

Short Answer

Step by step solution

Property of Polynomial:

Polynomial Operations:

Polynomial Operations:

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Question:Show that x2+2 is irreducible in Z5x .Factorx4-4 as a product of irreducible inZ5x .

Short Answer

Step by step solution

Property of Polynomial:

Polynomial Operations:

Polynomial Operations:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Theoretical and Mathematical Physics

Decision Maths

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

Question:
Show that $x^{2} + 2$ is irreducible in $Z_{5} [x]$ .
Factor $x^{4} - 4$ as a product of irreducible in $Z_{5} [x]$ .