What is minimal polynomial: Definition and 42 Discussions
In field theory, a branch of mathematics, the minimal polynomial of an element α of a field extension is, roughly speaking, the polynomial of lowest degree having coefficients in the field, such that α is a root of the polynomial. If the minimal polynomial of α exists, it is unique. The coefficient of the highestdegree term in the polynomial is required to be 1.
More formally, a minimal polynomial is defined relative to a field extension E/F and an element of the extension field E/F. The minimal polynomial of an element, if it exists, is a member of F[x], the ring of polynomials in the variable x with coefficients in F. Given an element α of E, let Jα be the set of all polynomials f(x) in F[x] such that f(α) = 0. The element α is called a root or zero of each polynomial in Jα
More specifically, Jα is the kernel of the ring homomorphism from F[x] to E which sends polynomials g to their value g(α) at the element α. Because it is the kernel of a ring homomorphism, Jα is an ideal of the polynomial ring F[x]: it is closed under polynomial addition and subtraction (hence containing the zero polynomial), as well as under multiplication by elements of F (which is scalar multiplication if F[x] is regarded as a vector space over F).
The zero polynomial, all of whose coefficients are 0, is in every Jα since 0αi = 0 for all α and i. This makes the zero polynomial useless for classifying different values of α into types, so it is excepted. If there are any nonzero polynomials in Jα, i.e. if the latter is not the zero ideal, then α is called an algebraic element over F, and there exists a monic polynomial of least degree in Jα. This is the minimal polynomial of α with respect to E/F. It is unique and irreducible over F. If the zero polynomial is the only member of Jα, then α is called a transcendental element over F and has no minimal polynomial with respect to E/F.
Minimal polynomials are useful for constructing and analyzing field extensions. When α is algebraic with minimal polynomial f(x), the smallest field that contains both F and α is isomorphic to the quotient ring F[x]/⟨f(x)⟩, where ⟨f(x)⟩ is the ideal of F[x] generated by f(x). Minimal polynomials are also used to define conjugate elements.
The usual theorem is talking about the linear operator being restricted to an invariant subspace:
I had no problem understanding its proof, it appears here for example: https://math.stackexchange.com/questions/3386595/restrictionoperatortwisdiagonalizableiftisdiagonalizable However, I...
I thought i understood the theorem below:
i) If A is a matrix in ##M_n(k)## and the minimal polynomial of A is irreducible, then ##K = \{p(A): p (x) \in k [x]\}## is a finite field
Then this example came up:
The polynomial ##q(x) = x^2 + 1## is irreducible over the real numbers and the matrix...
Hello! (Wave)
If the matrix $A \in M_n(\mathbb{C})$ has $m_A(x)=(x^2+1)(x^21)$ as its minimal polynomial, then I want to find the minimal polynomials of the matrices $A^2$ and $A^3$.
($M_n(k)$=the $n \times n$ matrices with elements over the field $k=\mathbb{R}$ or $k=\mathbb{C}$)
Is there a...
Homework Statement
Let a = (1+(3)^1/2)^1/2. Find the minimal polynomial of a over Q.
Homework EquationsThe Attempt at a Solution
Maybe the first thing to realize is that Q(a):Q is probably going to be 4, in order to get rid of both of the square roots in the expression. I also suspect that...
Homework Statement
So I need the find the minimal polynomial of the primitive 15th root of unity. Let's call this minimal polynomial m(x)
Homework EquationsThe Attempt at a Solution
I know that m(x) is an irreducible factor of x^15  1 and also that the degree of m(x) is equal to the Euler...
Homework Statement
Find the minimal polynomial of a = i*(2)^1/2 + (3)^1/2
Homework EquationsThe Attempt at a Solution
Well, I know the minimal polynomial will have degree four, and that's about it. Will it help if I look at the linear factors of the minimal polynomial in some splitting field...
Hey! :o
We suppose that $M/L/K$ are consecutive fields extensions and $a\in M$ is algebraic over $K$. I want to show that $a$ is algebraic also over $L$. I want to show also that the minimal polynomial of $a$ over $L$ divides the minimal polynomial of $a$ over $K$ (if we consider this...
I started by setting $\alpha= e^{2\pi i/3} + \sqrt[3]{2}.$ Then I obtained $f(x) = x^9  9x^6  27x^3  27$ has $\alpha$ as a root.
How can I proceed to find the minimal polynomial of $\alpha$ over $\mathbb{Q},$ and identify its other roots?
I'm trying find the minimal polynomial of a=3^{1/3}+9^{1/3} over the rational numbers. I am currently going about this by trying to construct a polynomial from a (using what I intuitively feel would be a sufficiently small number of operations).
Then I'd show it's irreducible by decomposing it...
Homework Statement .
Let ##X:=\{A \in \mathbb C^{n\times n} : rank(A)=1\}##. Determine a representative for each equivalence class, for the equivalence relation "similarity" in ##X##.
The attempt at a solution.
I am a pretty lost with this problem: I know that, thinking in terms of...
Homework Statement
M: V > V linear operator st M^2 + 1_v = 0
find the POSSIBILITIES for min. pol. of M^3+2M^2+M+3I_v
Homework Equations
The Attempt at a Solution
using M^2 = 1_v,
i rewrote the operator(?) as
M^3 + M + I_v
i don't know what to do. i guessed min poly to...
i understand how to find minimal poly. if a matrix is given. i am curious if you can find the matrix representation if minimal polynomial is given.
i'm not exactly sure how you could since you can possibly lose repeated evalues when you write minimal polynomial. how can u create a n...
Hi everyone, :)
This is one of the thoughts that I got after thinking about finding the minimal polynomial of a matrix. I know that the minimal polynomial is easy to find when a matrix is diagonalizable. Then the minimal polynomial only consist all the distinct linear factors of the...
Homework Statement
Given the matrix
2 0 0 0 0 0 0
1 2 0 0 0 0 0
0 1 2 0 0 0 0
0 0 1 2 0 0 0
0 0 0 0 2 0 0
0 0 0 0 1 2 0
0 0 0 0 0 0 2
What is the minimal polynomial?
Homework Equations

The Attempt at a Solution
This is the Jordan form, so I guess the solution is just...
Homework Statement
A matrix A\inMn(ℂ) is diagonalizable if and only if mA(x) has no repeated roots.
Homework Equations
If A\inHom(V,V) = {A:V→V  A is a linear map}, the minimal polynomial of A, mA(x), is the smallest degree monic polynomial f(x) such that f(A)=0.
The Attempt at a...
Let $L$ be an extension of a field $F$. Let $\alpha_1, \alpha_2\in L$ be such that both of them are algebraic over $F$ and have the same minimal polynomial $m$ over $F$. We know that there is an isomorphism $\phi:F(\alpha_1)\to F(\alpha_2)$ defined as $\phi(\alpha_1)=\alpha_2$ and $\phi(x)=x$...
Assume that \sqrt[M]{N} is irrational where N,M are positive integers. I'm under belief that
X^MN
is the minimal polynomial of \sqrt[M]{N} in \mathbb{Q}[X], but I cannot figure out the proof. Assume as an antithesis that p(X)\in\mathbb{Q}[X] is the minimal polynomial such that \partial p <...
Homework Statement
Given field extension C of Q, Find the minimal polynomial of a=sqrt( 5 + sqrt(23) ) (element of C).Homework Equations
The Attempt at a Solution
I may be complicating things, but let me know if you see something missing.
Doing the appropriate algebra, I manipulated the above...
Let A ∈ Mn×n(F )
Why dim span(In, A, A2, A3, . . .) = deg(mA)?? where mA is the minimal polynomial of A.
For span (In,A,A2...)
I can prove its
dimension <= n by CH Theorem
but what's the relation between
dim span(In,A,A2...)and deg(mA)
I've been given a matrix A and calculated the characteristic polynomial. Which is (1λ)5. Given this how does one calculate the minimal polynomial?
Also just to check, is it correct that the minimal polynomial is the monic polynomial with lowest degree that satisfies M(A)=0 and that all the...
Homework Statement
If we have a transformation matrix \begin{bmatrix} 1 & 2 & 4 \\0 & 0 & 0 \\0 & 0 & 0 \end{bmatrix}
Homework Equations
The Attempt at a Solution
I found the characteristic polynomial of this matrix: x^3  x^2 = x^2(x1) ...can anybody please help me...
Homework Statement
Let A = \begin{pmatrix}1 & 1 & 0 & 0\\1 & 1 & 0 & 0\\2 & 2 & 2 & 1\\ 1 & 1 & 1 & 0 \end{pmatrix}
The characteristic polynomial is f(x)=x^2(x1)^2. Show that f(x) is also the minimal polynomial of A.
Method 1: Find v having degree 4.
Method 2: Find a vector v of...
Homework Statement
Suppose that A is a 6x6 matrix with real values and has a min. poly of p(s) = s^3.
a) Find the Characteristic polynomial of A
b) What are the possibilities for the Jordan form of A?
c) What are the possibilities of the rank of A?
Homework Equations
See below...
Homework Statement
A n × nmatrix A satisfies the equation A2 = A.
(a) List all possible characteristic polynomials of A.
(b) Show that A is similar to a diagonal matrix
Homework Equations
The Attempt at a Solution
A2 = A
so, A2  A = 0
A(AI) = 0
Our minimal polynomial is x2  x = m(x)...
Homework Statement
Find the minimal polynomial of a=y^3 in F=Kron(Z/2Z, x^4+x+1). (Calculate the powers of a^2, a^3, and a^4.)
Homework Equations
The Attempt at a Solution
I attempted this trying to follow a similar worked problem in my book:
a=y^3 & y^4=y+1
Multiply by...
1.Let F=K(u) where u is transcedental over the field K. If E is a field such that K contained in E contained in F, then Show that u is algebraic over E.
Let a
be any element of E that is not in K. Then a = f(u)/g(u)
for some polynomials f(x), g(x) inK[x]
2.Let K contained in E...
1. Homework Statement [/b]
Let V be the space of all polynomials of degree less than or equal to 2 over the reals. Define the transformation, H, as a mapping from V to R[x] by (Hp)(x)=\int^x_{1}p(t)dt\\. a) Find the monic generator, d, which generates the ideal, M, containing the range of H...
Homework Statement
Let V be the vector space of n x n matrices over the field F. Fix A \in V. Let T be the linear operator on V defined by T(B) = AB, for all B \in V.
a). Show that the minimal polynomial for T equals the minimal polynomial for A.
b) Find the matrix of T with respect...
About the invariance of similar linear operators and their minimal polynomial
Notations:
F denotes a field
V denotes a vector space over F
L(V) denotes a vector space whose members are linear operators from V to V itself and its field is F, then L(V) is an algebra where multiplication is...
I'm just learning a bit about the "minimal polynomial" today but there was a section from the book which I didn't understand. This is the section, and I've circled the bit I'm having trouble with.
http://img15.imageshack.us/img15/1825/97503873.jpg [Broken] (sorry, it won't let me post an...
Homework Statement
a) Prove that if a polynomial f(lambda) has f(A)=0, then f(AT)=0
b) Prove that A and AT have the same minimal polynomial.
c) If A has a cyclic vector, prove that AT is similar to A.
2. The attempt at a solution
a) I know that I need to show that f(AT) =...
Find the minimal polynomial with root 21/3 + 21/2.
I would just use maple but I do not have it installed on this machine.
I found the polynomial and verified that this is indeed a root. I only have Eisenstiens criterion for determining whether it is irreducible, and I can not apply it in...
Here is my problem:
Let A be a complex n x n matrix with minimal polynomial q(x)=the sum from j=0 to m of \alpha_j x^j where m\leq n and \alpha_m = 1.
Show: If A is nonsingular then \alpha_0 does not equal 0.
So, I get that 0=q(A)=\alpha_0 I_n + \alpha_1 A + \alpha_2 A^2 +...+A^m...
Homework Statement
Let f(x) be an irreducible polynomial cubic in Q. For example
f(x) = ax^3 + bx^2 + cx + d
Let A be a 3 x 3 matrix with entries in Q such that char(A,x) = f(x). Find the minimal polynomial m(x) of A. Can you generalize to a degree n polynomial?
Homework Equations...
Homework Statement
Find the minimal polynomial of \frac{\sqrt{3}}{1+2^{1/3}} over Q
we'll call this x
Homework Equations
I wish I knew some :(
The Attempt at a SolutionBy taking powers of x, I was able to show that the extension Q(x) has degree six (since 21/3 and sqrt(3) are both...
Let A be an n x n matrix; denote its distinct eigenvalues by a_1,...,a_k and denote the index of a_i by d_i. How do I prove that the minimal polynomial is then:
m_A(s) = (sa_1)^d_1*...*(sa_k)^d_k
?
The characterstic polynomial is defined as:
p_A(s) = (sa_1)*...*(sa_n);
I have matrix A
\left(\begin{array}{ccc}6&2&2\\2&2&2\\2&2&2\end{array} \right)
Its characteristic polynomial is
p(\lambda)=\lambda^3  10\lambda^2 + 32\lambda 32
Finding minimal polynomial i get...
Given a matrix A how can I found its minimal polynomial? I know how to find its characteristic polynomial, but how do I reduce it to minimal?
Thanks,
Chen