- Let p∈ Fand suppose that pis the minimum polynomial for a ∈ F. Consider a field extension of F called G. Thus a ∈ G also. Show that the minimum polynomial of a with coefficients in G must divide p.
- Here is a polynomial in ℚ
Show it is irreducible in ℚ

. Now consider x^{2}− x + 1. Show that in ℚ∕it follows that≠0. Find its inverse in ℚ∕. - Here is a polynomial in ℚ
Show it is irreducible in ℚ

. Now consider x + 2. Show that in ℚ∕it follows that≠0. Find its inverse in ℚ∕. - Here is a polynomial in ℤ
_{3}Show it is irreducible in ℤ

_{3}. Showis not zero in ℤ_{3}∕. Now find its inverse in ℤ_{3}∕. - Suppose the degree of pis r where pis an irreducible monic polynomial with coefficients in a field F. It was shown that the dimension of F∕is r and that a basis is. Now let A be an r × r matrix and let q
_{i}= ∑_{k=1}^{r}A_{ij}x^{j−1}. Show thatis a basis for F∕if and only if the matrix A is invertible. - Suppose you have W a subspace of a finite dimensional vector space V . Suppose also that
dim= dim. Tell why W = V.
- Suppose V is a vector space with field of scalars F. Let T ∈ℒ, the space of linear transformations mapping V onto W where W is another vector space (See Problem 23 on Page 157.). Define an equivalence relation on V as follows. v ∼ w means v − w ∈ ker. Recall that ker≡. Show this is an equivalence relation. Now foran equivalence class define T
^{′}≡ Tv. Show this is well defined. Also show that with the operationsis a vector space. Show next that T^{′}: V∕ker→ W is one to one. This new vector space, V∕keris called a quotient space. Show its dimension equals the difference between the dimension of V and the dimension of ker. - ↑Suppose now that W = T. Then show that T
^{′}in the above is one to one and onto. Explain why dim= dim. Now see Problem 25 on Page 157. Show that - Let V be an n dimensional vector space and let W be a subspace. Generalize the Problem 7 to define and give properties of V∕W. What is its dimension? What is a basis?
- A number is transcendental if it is not the root of any nonzero polynomial with rational coefficients.
As mentioned, there are many known transcendental numbers. Suppose α is a real transcendental
number. Show that is a linearly independent set of real numbers if the field of scalars is the rational numbers.
- Suppose F is a countable field and let A be the algebraic numbers, those numbers in G which are
roots of a polynomial in F. Show A is also countable.
- This problem is on partial fractions. Suppose you have
where the polynomials q

_{i}are relatively prime and all the polynomials pand q_{i}have coefficients in a field of scalars F. Thus there exist polynomials a_{i}having coefficients in F such thatExplain why

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

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

where the degree of each r

_{i}is less than the degree of q_{i}and Mis a polynomial. Now argue that M= 0 . From this explain why the usual partial fractions expansion of calculus must be true. You can use the fact that every polynomial having real coefficients factors into a product of irreducible quadratic polynomials and linear polynomials having real coefficients. This follows from the fundamental theorem of algebra. - It was shown in the chapter that A is a field. Here A are the numbers in ℝ which are roots of a rational polynomial. Then it was shown in Problem 11 that it was actually countable. Show that A + iA is also an example of a countable field.

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