Name:
Due: Thursday, January 30.
Review Sections 29, 30 in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment in at the beginning of class on Thursday, January 30.
Remember to fully justify all your answers, and provide complete details.
1. Let
α =
Q
.)
√
3
+
√
5. Find f ( x ) ∈
Q
[ x ] such that f (
α
)
=
0. (This shows that
α is algebraic over
2. Find irr(
α, as irr(
α,
Q
Q
) and deg(
α,
Q
) for
α
) is actually irreducible.
=
√
3
+
2 i . You must prove that the polynomial you propose
3. Is
π 2 algebraic or transcendental over
Q
(
π 3 )?
4.
a.
Read Example 29.19 very carefully.
b.
Show that the polynomial x
2 +
1 is irreducible in
Z
3
[ x ].
c.
Let
α be a root of x
2 +
1 in an extension field of
Z 3
. Imitate Example 29.19, and give a multiplication table for the field
Z
3
(
α
).
Hint: This is a field with nine elements:
0
,
1
,
2
, α,
2
α,
1
+ α,
1
+
2
α,
2
+ α,
2
+
2
α
.
5. Let F be a field. Define F n =
{ ( a
1
, . . . , a n
) | a i
∈ F addition and scalar multiplication defined follows:
} . Show that F n is an F -vector space, where
( a
1
, . . . , a n
)
+
( b
1
, . . . , b n
)
=
( a
1
+ b
1
, . . . , a n
+ b n
) ;
λ
( a
1
, . . . , a n
)
=
(
λ a
1
, . . . , λ a n
)
.
6. Exercise 30.4.
7. Exercise 30.10.
8. Let V and V
0 be F -vector spaces. A function
φ
: V → V
0 is called a linear transformation if for all
α, β
∈ V and a ∈ F , we have
φ
(
α + β
)
= φ
(
α
)
+ φ
(
β
) ;
φ
( a
α
)
= a
φ
(
α
)
.
If
φ is also a bijection, we say that V and V 0 are isomorphic . Show that if V is an F -vector space of dimension n , then V is isomorphic to F n
.
9. Let V be a vector space over a field F .
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a.
Define what it means for a subset W ⊆ V to be subspace of the vector space V over F .
b.
Prove that an intersection of subspaces of V is again a subspace of V .
Through the course of this assignment, I have followed the Aggie
Code of Honor.
An Aggie does not lie, cheat or steal or tolerate those who do.
Signed:
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