Name:
Problem Set 2
Math 416, Section 200, Spring 2014
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, Q) and deg(α, Q) for α =
as irr(α, Q) is actually irreducible.
3.
√
3 + 2i. You must prove that the polynomial you propose
a. Read Example 29.19 very carefully.
b. Show that the polynomial x2 + 1 is irreducible in Z3 [x].
c. Let α be a root of x2 + 1 in an extension field of Z3 . Imitate Example 29.19, and give a
multiplication table for the field Z3 (α). Hint: This is a finite field. How many elements
does it have? Write those elements in terms of α. For instance, 0, 1, 2, α, 1 + α are all
elements of Z3 (α), but there are more.
4. Let E be an extension field of a finite field F, where F has q elements. Let α ∈ E algebraic
over F of degree n. Prove that F(α) has qn elements.
5. Let F be a field. Define F n = {(a1 , . . . , an ) | ai ∈ F}. Show that F n is an F-vector space, where
addition and scalar multiplication defined follows:
(a1 , . . . , an ) + (b1 , . . . , bn ) = (a1 + b1 , . . . , an + bn ) ;
λ(a1 , . . . , an ) = (λa1 , . . . , λan ).
6. Read the proof of Theorem 30.17.
State Theorem 30.17 and explain its proof in your own words.
7. Read the proof of Theorem 30.19.
State Theorem 30.19 and explain its proof in your own words.
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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(aα) = aφ(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.
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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