of F(a) tha 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.] fiold of 8 elements: of 16 elements; of 25 elements.

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Chapter2: Second-order Linear Odes
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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.
Transcribed Image Text: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.
Transcribed Image Text: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+b23+c232 | a,b,c is a subfield of .

We know that  is a proper subfield of .

Consider the polynomial x3-2 on the field .

Find the roots of the polynomial:

x3-2=0x3=2x=23

The root of the polynomial is x=23. Notice that 23 is not a rational number.

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