74 CHAPTER 3. VECTOR SPACES

an old problem called squaring the circle which was to construct a square with the samearea as a circle using a straight edge and compass. Such numbers are all algebraic. Thusthe fact π is transcendental implies this problem is impossible.2

3.5 Exercises1. Let p(x) ∈ F [x] and suppose that p(x) is the minimum polynomial for a ∈ F. Con-

sider a field extension of F called G. Thus a ∈ G also. Show that the minimumpolynomial of a with coefficients in G must divide p(x).

2. Here is a polynomial inQ [x] : x2+x+3. Show it is irreducible inQ [x]. Now considerx2− x+1. Show that in Q [x]/

(x2 + x+3

)it follows that

[x2− x+1

]̸= 0. Find its

inverse in Q [x]/(x2 + x+3

)3. Here is a polynomial inQ [x] : x2−x+2. Show it is irreducible inQ [x]. Now consider

x+2. Show that in Q [x]/(x2− x+2

)it follows that [x+2] ̸= 0. Find its inverse in

Q [x]/(x2− x+2

).

4. Here is a polynomial in Z3 [x] : x2 + x+ 2̄. Show it is irreducible in Z3 [x]. Show[x+ 2̄

]is not zero in Z3 [x]/

(x2 + x+ 2̄

). Now find its inverse in Z3 [x]/

(x2 + x+ 2̄

).

5. Suppose the degree of p(x) is r where p(x) is an irreducible monic polynomialwith coefficients in a field F. It was shown that the dimension of F [x]/(p(x)) isr and that a basis is

{1, [x] ,

[x2], · · · ,

[xr−1

]}. Now let A be an r× r matrix and let

qi (x) = ∑rk=1 Ai jx j−1. Show that {[q1 (x)] , · · · , [qr (x)]} is a basis for F [x]/(p(x)) if

and only if the matrix A is invertible.

6. Suppose you have W a subspace of a finite dimensional vector space V . Supposealso that dim(W ) = dim(V ) . Tell why W =V.

7. Suppose V is a vector space with field of scalars F. Let T ∈ L (V,W ) , the spaceof linear transformations mapping V onto W where W is another vector space (SeeProblem 23 on Page 62.). Define an equivalence relation on V as follows. v ∼wmeans v−w ∈ ker(T ) . Recall that ker(T )≡ {v : Tv = 0}. Show this is an equiv-alence relation. Now for [v] an equivalence class define T ′ [v] ≡ Tv. Show this iswell defined. Also show that with the operations

[v]+ [w]≡ [v+w] , α [v]≡ [αv]

this set of equivalence classes, denoted by V/ker(T ) is a vector space. Show nextthat T ′ : V/ker(T )→W is one to one. This new vector space, V/ker(T ) is called aquotient space. Show its dimension equals the difference between the dimension ofV and the dimension of ker(T ).

8. ↑Suppose now that W = T (V ) . Then show that T ′ in the above is one to one andonto. Explain why dim(V/ker(T )) = dim(T (V )) . Now see Problem 25 on Page62. Show that rank(T )+null(T ) = dim(V ) .

2Gilbert, the librettist of the Savoy operas, may have heard about this great achievement. In Princess Ida whichopened in 1884 he has the following lines. “As for fashion they forswear it, so they say - so they say; and the circle- they will square it some fine day some fine day.” Of course it had been proved impossible to do this a couple ofyears before.