Show that Z,[V3] = {a + bV3| a, b E Z,} is a field. For anypositive integer k and any prime p, determine a necessary and suf-ficient condition for Z,[Vk] = {a + bVk|a, b E Z,} to be a field.

We need to show that Z73=a+b3 a,b∈Z7 is a field. To show that Z7[3] is a field, we need to verify…

Answered: Show that Z,[V3] = {a + bV3| a, b E Z,}…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.3: The Characteristic Of A Ring

Problem 18E

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Question

Show that Z,[V3] = {a + bV3| a, b E Z,} is a field. For any
positive integer k and any prime p, determine a necessary and suf-
ficient condition for Z,[Vk] = {a + bVk|a, b E Z,} to be a field.

Expert Solution

Step 1

We need to show that $Z_{7} [\sqrt{3}] = \{a + b \sqrt{3} a, b \in Z_{7}\}$ is a field.

To show that $Z_{7} [\sqrt{3}]$ is a field, we need to verify the following:

$Z_{7} [\sqrt{3}]$ is a commutative ring with identity.
Every nonzero element in $Z_{7} [\sqrt{3}]$ has a multiplicative inverse.
To show that $Z_{7} [\sqrt{3}]$ is a commutative ring with identity, we need to show that it satisfies the following axioms:

Addition is commutative and associative.
There exists an additive identity element 0.
Every element has an additive inverse.
Multiplication is commutative and associative.
There exists a multiplicative identity element 1.
Every nonzero element has a multiplicative inverse. It is straightforward to verify that $Z_{7} [\sqrt{3}]$ satisfies all these axioms.

Step 2

To show that every nonzero element in $Z_{7} [\sqrt{3}]$ has a multiplicative inverse, we need to show that for any nonzero element $a + b \sqrt{3}$ in $Z_{7} [\sqrt{3}]$ , there exists an element $c + d \sqrt{3}$ in $Z_{7} [\sqrt{3}]$ such that $(a + b \sqrt{3}) (c + d \sqrt{3}) = 1$ . This can be done by solving the equation $(a + b \sqrt{3}) (c + d \sqrt{3}) = 1$ for c and d. We get $c = (\frac{a}{4}) + (\frac{3 b}{4})$ and $d = (\frac{3 a}{4}) + (\frac{b}{4})$ , where the division is taken in $Z_{7}$ . It can be verified that these values of c and d satisfy the equation, and hence every nonzero element in $Z_{7} [\sqrt{3}]$ has a multiplicative inverse.

Therefore, $Z_{7} [\sqrt{3}]$ is a field.

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Show that Z,[V3] = {a + bV3| a, b E Z,} is a field. For any positive integer k and any prime p, determine a necessary and suf- ficient condition for Z,[Vk] = {a + bVk|a, b E Z,} to be a field.