Section 29 number 34 33. Let E be an extension field of a field F and let a e E be transcendental over F. Show that every elementof F(a) that is not in F is also transcendental over F.34. Show that {a + b(/2) + c(2)2 | a, b, c e Q} is a subfield of R by using the ideas of this section, rather thanby a formal verification of the field axioms. [Hint: Use Theorem 29.18.]35. Following the idea of Exercise 31, show that there exists a field of 8 elements; of 16 elements; of 25 elements.t ouory element of F is algebraic over the prime field Z, < F. 29.18 TheoremLet E be a simple extension F(a) of a field F, and let a be algebraic over F. Letthe degree of irr(a, F) be n > 1. Then every element B of E = F(@) can be uniquelyexpressed in the formB = bo + bja ++ bn-10"-1,where the b; are in F.

Name: Section 29 number 34 33. Let E be an extension field of a field F and
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: Section 29 number 34 33. Let E be an extension field of a field F and let a e E be transcendental over F. Show that every elementof F(a) that is not in F is also transcendental over F.34. Show that {a + b(/2) + c(2)2 | a, b, c e Q} is a subfield of R

Prove the set a+b23+c232 | a,b,c∈ℚ is a subfield of ℝ. We know that ℚ is a proper subfield of ℝ.…

Answered: of F(a) tha 34. Show that {a + b(2) +…

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

Concept explainers

Topic Video

Question

Section 29 number 34

33. Let E be an extension field of a field F and let a e E be transcendental over F. Show that every element
of F(a) that is not in F is also transcendental over F.
34. Show that {a + b(/2) + c(2)2 | a, b, c e Q} is a subfield of R by using the ideas of this section, rather than
by a formal verification of the field axioms. [Hint: Use Theorem 29.18.]
35. Following the idea of Exercise 31, show that there exists a field of 8 elements; of 16 elements; of 25 elements.
t ouory element of F is algebraic over the prime field Z, < F.

29.18 Theorem
Let E be a simple extension F(a) of a field F, and let a be algebraic over F. Let
the degree of irr(a, F) be n > 1. Then every element B of E = F(@) can be uniquely
expressed in the form
B = bo + bja +
+ bn-10"-1,
where the b; are in F.

Expert Solution

Step 1

Prove the set $\{a + b (\sqrt[3]{2}) + c {(\sqrt[3]{2})}^{2} | a, b, c \in ℚ\}$ is a subfield of $ℝ$ .

We know that $ℚ$ is a proper subfield of $ℝ$ .

Consider the polynomial $x^{3} - 2$ on the field $ℚ$ .

Find the roots of the polynomial:

$x^{3} - 2 = 0 x^{3} = 2 x = \sqrt[3]{2}$

The root of the polynomial is $x = \sqrt[3]{2}$ . Notice that $\sqrt[3]{2}$ is not a rational number.

Trending now

This is a popular solution!

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Discrete Probability Distributions$

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.

Similar questions