7.4. EXERCISES 213

40. As mentioned, for distinct algebraic numbers αi, the complex numbers {eαi}ni=1 arelinearly independent over the field of scalars A where A denotes the algebraic numbers,those which are roots of a polynomial having integer (rational) coefficients. What isthe dimension of the vector space C with field of scalars A, finite or infinite? If thefield of scalars were C instead of A, would this change? What if the field of scalarswere R?

41. Suppose F is a countable field and let A be the algebraic numbers, those numbers inG which are roots of a polynomial in F [x]. Show A is also countable.

42. This problem is on partial fractions. Suppose you have

R (x) =p (x)

q1 (x) · · · qm (x), degree of p (x) < degree of denominator.

where the polynomials qi (x) are relatively prime and all the polynomials p (x) andqi (x) have coefficients in a field of scalars F. Thus there exist polynomials ai (x)having coefficients in F such that

1 =

m∑i=1

ai (x) qi (x)

Explain why

R (x) =p (x)

∑mi=1 ai (x) qi (x)

q1 (x) · · · qm (x)=

m∑i=1

ai (x) p (x)∏j ̸=i qj (x)

Now continue doing this on each term in the above sum till finally you obtain anexpression of the form

m∑i=1

bi (x)

qi (x)

Using the Euclidean algorithm for polynomials, explain why the above is of the form

M (x) +

m∑i=1

ri (x)

qi (x)

where the degree of each ri (x) is less than the degree of qi (x) and M (x) is a poly-nomial. Now argue that M (x) = 0. From this explain why the usual partial fractionsexpansion of calculus must be true. You can use the fact that every polynomial havingreal coefficients factors into a product of irreducible quadratic polynomials and linearpolynomials having real coefficients. This follows from the fundamental theorem ofalgebra in the appendix.

43. Suppose {f1, · · · , fn} is an independent set of smooth functions defined on some inter-val (a, b). Now let A be an invertible n×n matrix. Define new functions {g1, · · · , gn}as follows. 

g1...

gn

 = A

f1...

fn

Is it the case that {g1, · · · , gn} is also independent? Explain why.

44. A number is transcendental if it is not the root of any nonzero polynomial with rationalcoefficients. As mentioned, there are many known transcendental numbers. Supposeα is a real transcendental number. Show that

{1, α, α2, · · ·

}is a linearly independent

set of real numbers if the field of scalars is the rational numbers.