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Abstract Algebra: An Introduction

Found in: Page 94

Book edition 3rd

Author(s) Thomas W Hungerford, David Leep

Pages 608 pages

ISBN 9781111569624

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Short Answer

Show that 1+3x is a unit in $ℤ_{9} [x]$ . Hence, Corollary 4.5 may be false if R is not an integral domain.

It is proved that 1+3x is a unit in $ℤ_{9} [x]$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Polynomial Arithmetic

If any given function R[x] is a ring, then the commutative, associative, and distributive laws hold such that the function f(x)+g(x) exists.

Step 2: Fields:

The given unit is 1+3x . The multiplicative inverse of this will be: 1-3x.

Then, we have:

$(1 + 3 x) (1 - 3 x) = 1 - 9 x^{2} = 1$

Clearly found to be in $ℤ_{9} [x]$ .

Hence proved, 1+3x is a unit in $ℤ_{9} [x]$ .

Most popular questions for Math Textbooks

Let $ℚ [\sqrt{2}]$ be the set of all real numbers of the form

$r_{0} + r_{1} \sqrt{2} + r_{2} {(\sqrt{2})}^{2} + \dots + r_{n} {(\sqrt{2})}^{n}$ , with $n \geq 0$ and $r_{i} \in ℚ$ .

Show that $ℚ [\sqrt{2}]$ is a subring of $ℝ$ .
Show that the function $θ : ℚ [x] \to ℚ [\sqrt{2}]$ defined by $θ (f (x)) = f (\sqrt{2})$ is a surjective homomorphism, but not an isomorphism. Thus Theorem 4.1 is not true with $R = ℚ$ and $\sqrt{2}$ in place of x. Compare this with Exercise 25.

If R is commutative, show that R[x] is also commutative.

Perform the indicated operation and simplify your answer:

$(3 x^{4} + 2 x^{3} - 4 x^{2} + x + 4) + (4 x^{3} + x^{2} + 4 x + 3) i n ℤ_{5}$
${(x + 1)}^{3} i n ℤ_{3} [x]$
${(x - 1)}^{5} i n ℤ_{5} [x]$
$(x^{2} - 3 x + 2) (2 x^{3} - 4 x + 1) i n ℤ_{7}$

Prove Theorem 4.10.

Find the gcd of $x + a + b$ and $x^{3} - 3 a b x + a^{3} + b^{3}$ in $ℚ [x]$ .

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