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Chapter 18: Problem 11

Let \(\tau \in \mathcal{L}_{F}(V)\) have non-zero minimal polynomial \(\phi\) of degree \(m,\) and let \(\phi=\phi_{1}^{e_{1}} \cdots \phi_{r}^{e_{r}}\) be the factorization of \(\phi\) into monic irreducible polynomials in \(F[X] .\) Let \(\odot\) be the scalar multiplication associated with \(\tau .\) Show that \(\beta \in V\) has minimal polynomial \(\phi\) under \(\tau\) if and only if \(\phi / \phi_{i} \odot \beta \neq 0\) for \(i=1, \ldots, r\)

Short Answer

Expert verified
Question: Show that a basis vector 尾 has a minimal polynomial 蠁 of degree m under a linear transformation 蟿 if and only if the factorization of 蠁 into monic irreducible polynomials does not annihilate 尾 when scalar multiplication associated with 蟿 is performed. Answer: A basis vector 尾 has a minimal polynomial 蠁 of degree m under a linear transformation 蟿 if and only if 蠁 / 蠁_i 鈯� 尾 鈮� 0 for all i = 1, ..., r, where 蠁 is factored into monic irreducible polynomials 蠁_1, ..., 蠁_r.

Step by step solution

01

Find the minimal polynomial of the linear transformation 蟿

Given that 蟿 鈭� 饾攺_F(V) has a non-zero minimal polynomial 蠁 of degree m. The minimal polynomial of a linear transformation is a polynomial that satisfies the condition: p(蟿)(尾) = 0, 鈭� 尾 鈭� V.
02

Recognize the minimal polynomial 蠁 of degree m and its factorization into monic irreducible polynomials

We are given that the minimal polynomial 蠁 factorizes as: 蠁 = 蠁_1^{e_1}...蠁_r^{e_r}, where 蠁_1, ..., 蠁_r are monic irreducible polynomials in F[X].
03

Establish the conditions that 蠁 / 蠁_i 鈯� 尾 鈮� 0 for i=1, ..., r

We need to show that the necessary and sufficient condition for 尾 to have minimal polynomial 蠁 under 蟿 is that: 蠁 / 蠁_i 鈯� 尾 鈮� 0, 鈭�i 鈭� {1, ..., r}.
04

Prove that 尾 has minimal polynomial 蠁 under 蟿 if the conditions in step 3 are satisfied

(鈬�): Suppose that 尾 has minimal polynomial 蠁 under 蟿. Then, we have 蠁(蟿)(尾) = 0. Now, let's consider the expression: 蠁 / 蠁_i 鈯� 尾 = (蠁_1^{e_1}...蠁_r^{e_r} / 蠁_i)(蟿)(尾) We must show this expression is not zero for all i 鈭� {1, ..., r}. If 蠁 / 蠁_i 鈯� 尾 = 0, then 蠁_i must divide 蠁. But, since each 蠁_i is irreducible and they are all distinct, they are coprime to each other, thus, 蠁 / 蠁_i 鈯� 尾 鈮� 0. (鈬�) Suppose that 蠁 / 蠁_i 鈯� 尾 鈮� 0 for all i 鈭� {1, ..., r}. For any polynomial g 鈭� F[X], if g(蟿)(尾) = 0, then g must be divisible by 蠁. Thus, 尾 has minimal polynomial 蠁 under 蟿. Since 蠁 is the smallest degree polynomial that satisfies the conditions mentioned, we conclude that 尾 has minimal polynomial 蠁 under 蟿 if and only if 蠁 / 蠁_i 鈯� 尾 鈮� 0 for i = 1, ..., r.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Transformation
A linear transformation, often denoted as \( \tau \), is a map between two vector spaces that preserves the operations of vector addition and scalar multiplication. In simpler terms, if you have vectors \( u \) and \( v \), and a scalar \( c \), then a linear transformation satisfies:
  • \( \tau(u + v) = \tau(u) + \tau(v) \)
  • \( \tau(c \cdot u) = c \cdot \tau(u) \)
These properties ensure that the transformation behaves nicely with respect to the vector space structure.
To understand this in the context of the original exercise, think of \( \tau \) acting on a vector space \( V \) over a field \( F \). The linear transformation \( \tau \) has a minimal polynomial \( \phi \) which is the simplest polynomial with coefficients in \( F \) that when applied to \( \tau \) results in the zero transformation. This polynomial gives insights into the characteristics and structure of the transformation \( \tau \).
Monic Irreducible Polynomials
Monic irreducible polynomials play a key role in the study of minimal polynomials. A polynomial is called monic if the leading coefficient (the coefficient of the highest power of \( X \)) is 1. Irreducibility means that the polynomial cannot be factored into polynomials of lower degree with coefficients in the same field.
For example, consider the polynomial \( X^2 + 1 \) over the real numbers. This polynomial is irreducible because it cannot be rewritten as a product of lower-degree polynomials with real coefficients. In contrast, \( X^2 - 1 \) can be factored as \( (X-1)(X+1) \), so it is not irreducible.
In the context of our original exercise, the minimal polynomial \( \phi \) is factored into monic irreducible polynomials as \( \phi = \phi_1^{e_1}\cdots\phi_r^{e_r} \). The condition \( \phi / \phi_i \odot \beta eq 0 \) leverages this factorization. It ensures that each irreducible component \( \phi_i \) captures unique information about the vector \( \beta \) and its interaction with the linear transformation \( \tau \).
Scalar Multiplication
Scalar multiplication is a fundamental operation in linear algebra, closely linked to vector spaces. Given a scalar \( a \) from a field \( F \) and a vector \( v \) from a vector space \( V \), the scalar multiplication \( a \cdot v \) results in another vector in \( V \).
Scalar multiplication has two essential properties:
  • It is commutative, i.e., \( a \cdot v = v \cdot a \).
  • It is distributive over both vector addition and scalar addition.
In our exercise, the notation \( \odot \) is used for scalar multiplication associated with the linear transformation \( \tau \). This extends the concept to elements like polynomials acting on vectors or transformations. When we encounter expressions like \( \phi / \phi_i \odot \beta \), it signifies not just multiplication of polynomials by a vector but how these polynomials as scalars interact within the transformation structure.
This operation is crucial because it helps identify if a vector \( \beta \) satisfies certain polynomial conditions under the linear transformation \( \tau \). When the expressions are non-zero, as required by the exercise, it confirms that \( \phi \), and ultimately the minimal polynomial, reflects the minimal algebraic dependencies of \( \beta \) under \( \tau \).

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Most popular questions from this chapter

Let \(\Psi=\left\\{\alpha_{i}\right\\}_{i=0}^{\infty}\) be a sequence of elements of an \(F\) -vector space \(V\). Further, suppose that \(\Psi\) has non- zero minimal polynomial \(\phi .\) (a) Show that for all polynomials \(g, h \in F[X],\) if \(g \equiv h(\bmod \phi),\) then \(g \star \Psi=h \star \Psi\). (b) Let \(m:=\operatorname{deg}(\phi) .\) Show that if \(g \in F[X]\) and \(\left(X^{i} g\right) \star \Psi=0\) for all \(i=0, \ldots, m-1,\) then \(g\) is a generating polynomial for \(\Psi\).

Let \(f \in F[X]\) be a monic polynomial of degree \(\ell>0\) over a field \(F\) (not necessarily finite), and let \(E:=F[X] /(f) .\) Further, suppose that \(f\) is irreducible, so that \(E\) is itself a field. Show how to compute the minimal polynomial of \(\alpha \in E\) over \(F\) deterministically, using algorithms that satisfy the following complexity bounds: (a) \(O\left(\ell^{3}\right)\) operations in \(F\) and space for \(O(\ell)\) elements of \(F\); (b) \(O\left(\ell^{2.5}\right)\) operations in \(F\) and space for \(O\left(\ell^{1.5}\right)\) elements of \(F\).

In this exercise, you are to derive the fundamental theorem of finite dimensional \(F[X]\) -modules, which is completely analogous to the fundamental theorem of finite abelian groups. Both of these results are really special cases of a more general decomposition theorem for modules over a principal ideal domain. Let \(V\) be an \(F[X]\) -module. Assume that as an \(F\) -vector space, \(V\) has finite dimension \(\ell>0,\) and that the \(F[X]\) -exponent of \(V\) is generated by the monic polynomial \(\phi \in F[X]\) (note that \(1 \leq \operatorname{deg}(\phi) \leq \ell\) ). Show that there exist monic, non- constant polynomials \(\phi_{1}, \ldots, \phi_{t} \in F[X]\) such that $$ \text { - } \phi_{i} \mid \phi_{i+1} \text { for } i=1, \ldots, t-1, \text { and } $$ \- \(V\) is isomorphic, as an \(F[X]\) -module, to the direct product of \(F[X]\) -modules $$ V^{\prime}:=F[X] /\left(\phi_{1}\right) \times \cdots \times F[X] /\left(\phi_{t}\right) $$ Moreover, show that the polynomials \(\phi_{1}, \ldots, \phi_{t}\) satisfying these conditions are uniquely determined, and that \(\phi_{t}=\phi .\) Hint: one can just mimic the proof of Theorem \(6.45,\) where the exponent of a group corresponds to the \(F[X]\) -exponent of an \(F[X]\) -module, and the order of a group element corresponds to the \(F[X]\) -order of an element of an \(F[X]\) -module - everything translates rather directly, with just a few minor, technical differences, and the previous exercise is useful in proving the uniqueness part of the theorem.

Let \(F\) be a finite field, and let \(V\) have finite dimension \(\ell>0\) over \(F\). Let \(\tau \in \mathcal{L}_{F}(V)\) have minimal polynomial \(\phi,\) with \(\operatorname{deg}(\phi)=m(\) and of course, by Theorem \(18.13,\) we have \(m \leq \ell) .\) Suppose that \(\alpha_{1}, \ldots, \alpha_{s}\) are randomly chosen elements of \(V\). Let \(g_{j}\) be the minimal polynomial of \(\alpha_{j}\) under \(\tau,\) for \(j=\) \(1, \ldots, s .\) Let \(Q\) be the probability that \(\operatorname{lcm}\left(g_{1}, \ldots, g_{s}\right)=\phi .\) The goal of this exercise is to show that \(Q \geq \Lambda_{F}^{\phi}(s),\) where \(\Lambda_{F}^{\phi}(s)\) is as defined in \(\S 18.3 .\) (a) Using Theorem 18.12 and Theorem \(18.11,\) show that if \(m=\ell,\) then \(Q=\) \(\Lambda_{F}^{\phi}(s)\) (b) Without the assumption that \(m=\ell,\) things are a bit more challenging. Adopting the matrix-oriented point of view discussed at the end of 818.3 , and transposing everything, show that \(-\) there exists \(\pi \in \mathcal{D}_{F}(V)\) such that the sequence \(\left\\{\pi \circ \tau^{i}\right\\}_{i=0}^{\infty}\) has minimal polynomial \(\phi,\) and \(-\) if, for \(j=1, \ldots, s,\) we define \(h_{j}\) to be the minimal polynomial of the sequence \(\left\\{\pi\left(\tau^{i}\left(\alpha_{j}\right)\right)\right\\}_{i=0}^{\infty},\) then the probability that \(\operatorname{lcm}\left(h_{1}, \ldots, h_{s}\right)=\) \(\phi\) is equal to \(\Lambda_{F}^{\phi}(s)\) (c) Show that \(h_{j} \mid g_{j},\) for \(j=1, \ldots, s,\) and conclude that \(Q \geq \Lambda_{F}^{\phi}(s)\).

If \(|F|=q\) and \(\phi \in F[X]\) is monic and factors into monic irreducible polynomials in \(F[X]\) as \(\phi=\phi_{1}^{e_{1}} \cdots \phi_{r}^{e_{r}},\) show that $$ \Lambda_{F}^{\phi}(1)=\prod_{i=1}^{r}\left(1-q^{-\operatorname{deg}\left(\phi_{i}\right)}\right) \geq 1-\sum_{i=1}^{r} q^{-\operatorname{deg}\left(\phi_{i}\right)} $$ From this, conclude that the probability that Algorithm MP terminates after just one loop iteration is \(1-O(m / q),\) where \(m=\operatorname{deg}(\phi) .\) Thus, if \(q\) is very large relative to \(m,\) it is highly likely that Algorithm MP terminates after just one iteration of the main loop.
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