How do I show 8.34? Could you explain this in great detail? I was told to use 8.33 to show 8.34 but I am not sure how to do this. I also attached list of definitions and theorems in textbook. **8.28.** For each of the following rings, which elements are units? Which are zero divisors?1. $\mathbb{Z}_{18}$2. $\mathbb{Z}_3 \oplus \mathbb{Z}_9$**8.32.** Let $R$ and $S$ be rings. Under precisely what circumstances is $R \oplus S$ an integral domain?**8.33.** Let $F$ be a field with subfields $K$ and $L$. Show that $K \cap L$ is a subfield of $F$. Extend this to show that the intersection of any collection of subfields is a subfield.**8.34.** Let $p$ be a prime and $F$ a field with $p^2$ elements. Show that $F$ cannot have more than one proper subfield. **Definition 8.8.** Let $ R $ be a commutative ring. Then a nonzero element $ a \in R $ is said to be a **zero divisor** if there exists a nonzero $ b \in R $ such that $ ab = 0 $.**Example 8.19.** In $ \mathbb{Z}_6 $, we note that 4 is a zero divisor, as $ 4 \cdot 3 = 0 $. On the other hand, 5 is not a zero divisor.**Example 8.20.** The ring of integers has no zero divisors.**Definition 8.9.** An **integral domain** is a commutative ring $ R $ with identity $ 1 \neq 0 $ having no zero divisors.**Example 8.21.** The rings $ \mathbb{Z}, \mathbb{Q}, \mathbb{R} $ and $ \mathbb{C} $ are all integral domains.**Example 8.22.** The polynomial ring $ \mathbb{R}[x] $ is an integral domain. Indeed, we know that it is a commutative ring with identity. Also, if $ f(x) = a_0 + a_1x + \cdots + a_nx^n $ and $ g(x) = b_0 + b_1x + \cdots + b_mx^m $, with $ a_i, b_i \in R $ and $ a_n \neq 0 \ne b_m $, then the unique term of highest degree in $ f(x)g(x) $ is $ a_nb_mx^{m+n} $. As $ R $ is an integral domain, $ a_nb_m \neq 0 $. Thus, $ f(x)g(x) $ is not the zero polynomial.**Example 8.23.** The rings $ \mathbb{Z}\mathbb{Z}, \mathbb{Z}_6 $ and $ M_2(\mathbb{R}) $ all fail to be integral domains. The first lacks an identity, the second has zero divisors and the third is not commutative.**Theorem 8.7 (Cancellation Law).** Let \( R \

Name: How do I show 8.34? Could you explain this in great detail? I was told
Uploaded: 2024-03-20T07:06:16.000Z
Channel: Bartleby
Description: How do I show 8.34? Could you explain this in great detail? I was told to use 8.33 to show 8.34 but I am not sure how to do this. I also attached list of definitions and theorems in textbook. **8.28.** For each of the following rings, which elements

Given that F is a Field with q=p2 elements. It is required to prove that F cannot have more than one proper subfield. Let us assume that F has more than one proper subfield Let K and L be two proper subfields of F. Then by 8.33, K∩L is also a subfield of F By a theorem (Subfield Criterion) note that, Let F be a Field with q=pnelements. Then every subfield of F has order pm, where m is a positive divisor of n. Conversely, if m is a positive divisor of n, then there is exactly one subfield of F with pm elements. Here n=2 1,2 are the positive divisors of 2 Then by the above theorem (Subfield Criterion ), every subfield of F has order p or p2. Since K and L are proper subfields of F, the subfields K and L has order p That is K and L has p elements. Note that , Any two fields having same number of elements are isomorphic. Thus K and L are isomorphic, K ≅ L This implies that any two proper subfields of F are isomorphic. Also, K∩L is a subfield of F and K∩L has order p then K∩L is isomorphic to K and L This implies that intersection of any two proper subfields of F are isomorphic to the subfields. Thus F has exactly one proper subfield. Hence F cannot have more than one proper subfield. Another method to prove this theorem: By a theorem (Subfield Criterion) note that , Let F be a Field with q=pnelements. Then every subfield of F has order pm, where m is a positive divisor of n. Conversely, if m is a positive divisor of n, then there is exactly one subfield of F with pm elements. Given that F is a Field with q=p2 elements Here n=2 1,2 are the positive divisors of 2 Then from the converse part of the above theorem (Subfield Criterion ), there is exactly one subfield of F with p elements and one subfield of F with p2 elements. Subfield of F with p elements is the only proper subfield of F. Thus there is exactly one proper subfield of F Hence F cannot have more than one proper subfield.

Answered: 8.34. Let p be a prime and F a field…

Math

Advanced Math

8.34. Let p be a prime and F a field with p² elements. Show that F cannot have more than one proper subfield.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: 2. (4 points) Verify that 223 is a prime number. Show your work and explain your reasoning.

Q: SSSx? +y? +z*dxdydz

A: We will solve the triple integration. First, we will integrate with respect to x . 2nd , we…

Q: Use descent to find x, y such that x2 + y? = 881 starting with 84² + 89² = 17 · 881.

Q: 2 F5V/² W

A: let f(w)=5w25

Q: n 1. A unit fraction is a fraction of the form where n is a positive integer. Note that the unit…

A: It is given that the unit fraction can be written as the sum of two unit fractions in the given…

Q: Can you explain how 0.1003 was obtained? because I do not understand how to locate 0.10 on the table

A: Given that, μ=19 daysσ=1 days The 10th percentile for incubation times is,

Q: Determine, with justification, whether the series converges absolutely, converges conditionally, or…

Q: iven P(A^c-complement)=0.40 P(B)=0.25 P(AUB)=0.75 P(A)=0.60 What is P((AnB)^c-complement) I…

A: Solution: Given information: P(AC)= 0.40P(B)=0.25P(A∪B)=0.75P(A)= 0.60

Q: I don't understand how the 2-adic representation of 526 is 1000011110. Please explain. Thank you

Q: 16. Write the perimeter of the rectangle in simplest form. 2v + 13 v-5

Q: What power of 9 is equivalent to (3) ?

A: To find the power of 9 that is equivalent to , we can set up the equation:To solve for x, we can…

Q: For all integers n, if 6 | n, then 9 | ((n^2) + 21n/2)

Q: Show that this number is Irrational number

Q: Which of the following is true when 2/3 +21²-31-3 is divided 1-5? Op (5)=282 Op (5) = 140 Op (5)=…

A: To determine the true statement when is divided by .

Q: Use descent to find x, y such that x? + y? = 881 starting with 842 + 892 = 17 · 881.

A: Partial derivative of a function with respect to x is computed by taking y as constant. Partial…

Q: Let 0 4-(19) and B-(1-1/2) A = 13 (1) Determine A2021

Q: The product of any four consecutive integers is divisible by 24.

A: We will use basic knowledge of integers and congruences to solve this question. The question says…

Concept explainers

Inverse Normal Distribution

The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.

Mean, Median, Mode

It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.

Z-Scores

A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.

Topic Video

Question

100%

How do I show 8.34? Could you explain this in great detail? I was told to use 8.33 to show 8.34 but I am not sure how to do this. I also attached list of definitions and theorems in textbook.

**8.28.** For each of the following rings, which elements are units? Which are zero divisors?
1. $\mathbb{Z}_{18}$
2. $\mathbb{Z}_3 \oplus \mathbb{Z}_9$

**8.32.** Let $R$ and $S$ be rings. Under precisely what circumstances is $R \oplus S$ an integral domain?

**8.33.** Let $F$ be a field with subfields $K$ and $L$. Show that $K \cap L$ is a subfield of $F$. Extend this to show that the intersection of any collection of subfields is a subfield.

**8.34.** Let $p$ be a prime and $F$ a field with $p^2$ elements. Show that $F$ cannot have more than one proper subfield.

**Definition 8.8.** Let $ R $ be a commutative ring. Then a nonzero element $ a \in R $ is said to be a **zero divisor** if there exists a nonzero $ b \in R $ such that $ ab = 0 $.

**Example 8.19.** In $ \mathbb{Z}_6 $, we note that 4 is a zero divisor, as $ 4 \cdot 3 = 0 $. On the other hand, 5 is not a zero divisor.

**Example 8.20.** The ring of integers has no zero divisors.

**Definition 8.9.** An **integral domain** is a commutative ring $ R $ with identity $ 1 \neq 0 $ having no zero divisors.

**Example 8.21.** The rings $ \mathbb{Z}, \mathbb{Q}, \mathbb{R} $ and $ \mathbb{C} $ are all integral domains.

**Example 8.22.** The polynomial ring $ \mathbb{R}[x] $ is an integral domain. Indeed, we know that it is a commutative ring with identity. Also, if $ f(x) = a_0 + a_1x + \cdots + a_nx^n $ and $ g(x) = b_0 + b_1x + \cdots + b_mx^m $, with $ a_i, b_i \in R $ and $ a_n \neq 0 \ne b_m $, then the unique term of highest degree in $ f(x)g(x) $ is $ a_nb_mx^{m+n} $. As $ R $ is an integral domain, $ a_nb_m \neq 0 $. Thus, $ f(x)g(x) $ is not the zero polynomial.

**Example 8.23.** The rings $ \mathbb{Z}\mathbb{Z}, \mathbb{Z}_6 $ and $ M_2(\mathbb{R}) $ all fail to be integral domains. The first lacks an identity, the second has zero divisors and the third is not commutative.

**Theorem 8.7 (Cancellation Law).** Let \( R \

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Centre, Spread, and Shape of a Distribution$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.