# 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 highest-degree 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 non-zero 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.

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1. ### I If T is diagonalizable then is restriction operator diagonalizable?

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/restriction-operator-t-w-is-diagonalizable-if-t-is-diagonalizable However, I...
2. ### A Computing the Minimal polynomial - Ring Theory

Am going through this notes...kindly let me know if there is a mistake on highlighted part. I think it ought to be; ##α^2=5+2\sqrt{6}##
3. ### I Finite fields, irreducible polynomial and minimal polynomial theorem

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...
4. ### MHB Minimal polynomial of matrices

Hello! (Wave) If the matrix $A \in M_n(\mathbb{C})$ has $m_A(x)=(x^2+1)(x^2-1)$ 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...
5. ### Finding the minimal polynomial of an irrational over Q

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...
6. ### Finding the minimal polynomial of primitive 15th root of 1

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...
7. ### Finding the minimal polynomial over Q

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...
8. ### MHB Algebraic element - Minimal polynomial

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...
18. ### Minimal polynomial of N^(1/M)

Assume that \sqrt[M]{N} is irrational where N,M are positive integers. I'm under belief that X^M-N 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 <...
19. ### Abstract Algebra- Finding the Minimal Polynomial

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...
20. ### The relation between span(In,A,A2, )and it's minimal polynomial

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)
21. ### Calculating the Minimal Polynomial for a Given Matrix A: A Guide

I've been given a matrix A and calculated the characteristic polynomial. Which is (1-λ)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...
22. ### How do you find the minimal polynomial?

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(x-1) ...can anybody please help me...
23. ### What is a minimal polynomial?

I have a small idea on what irreducible and primitive polynomials are in Abstract algebra. But what is minimal polynomial? -Devanand T
24. ### Show that Characteristic polynomial = minimal polynomial

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(x-1)^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...
25. ### Minimal Polynomial and Jordan Form

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...
26. ### Matrix, minimal polynomial

Homework Statement A n × n-matrix 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(A-I) = 0 Our minimal polynomial is x2 - x = m(x)...
27. ### Finding a Minimal Polynomial

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...
28. ### When p(A)=0 iff p(B)=0 for any polynomial,why same minimal polynomial?

For two matrices A and B, when p(A)=0 iff p(B)=0 for any polynomial, what will happen? i read that A and B have the same minimal polynomial, why?
29. ### Minimal Polynomial, Algebraic Extension

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...
30. ### Monic Generator (Minimal Polynomial)

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...
31. ### Minimal Polynomial Question

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...
32. ### About the invariance of similar linear operators and their minimal polynomial

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...
33. ### Understanding the Minimal Polynomial: Clarifying Confusion on p(T)(v)

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...
34. ### Minimal polynomial, transpose, similar

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) =...
35. ### Find the minimal polynomial with real root

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...
36. ### Show Non-Singular Matrix A Has Non-Zero Minimal Polynomial Coefficient

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 non-singular 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...
37. ### Finding the minimal polynomial of a matrix?

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...
38. ### Find the minimal polynomial

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...
39. ### Minimal Polynomial A nxn Matrix

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) = (s-a_1)^d_1*...*(s-a_k)^d_k ? The characterstic polynomial is defined as: p_A(s) = (s-a_1)*...*(s-a_n);
40. ### A question on minimal polynomial (LA)

f(t)=t^n+a_n-1t^(n-1)+...+a1t+a0 there's a square matrix of order n, A: \bordermatrix{ & & & & \cr 0 & 0 & ... & 0& -a_0 \cr 1 &0 & ... & 0 & -a_1 \cr ... & ... & ... & ... & ... \cr 0 & 0 & ... & 1 & -a_{n-1}\cr} show that f(t) is...
41. ### Explaining Minimal Polynomial of Matrix A

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...
42. ### Matrix Minimal Polynomial

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