Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [minimal-polynomials]

This is the lowest order monic polynomial satisfied by an object, such as a matrix or an algebraic element over a field.

For instance, $\sqrt2$ is an algebraic number, that is, it's a root of a non-zero polynomial with rational coefficients. Its minimal polynomial is $x^2-2$, since $\sqrt2$ is a root of this (monic) polynomial, and it is not a root of a non-zero polynomial with rational coefficients of a smaller degree.

1298 questions
0
votes
1 answer

Minimal polynomial over $\Bbb Q$ of an irrational number $\alpha$ with $\alpha^3 + 3\alpha^2-2=0$

I am revising for my exams and don't understand how to do the following question, any hints would be very helpful! Find a minimal polynomial of $\alpha$ when $\alpha$ is an irrational number satisfying $\alpha^3 + 3\alpha^2-2=0$. I have the…
Koala
  • 407
0
votes
1 answer

Show $f(x) = x^m$ for some positive integer m

new to the site and was stumped on this question, hoping for some help :) So I have an nxn matrix A lets say, with 0 as the only eigenvalue. And let $f(x)$ be the minimal polynomial of $A$ with a leading coefficient of $1$. I have to show: $f(x)=…
Bugcatcher123
  • 111
-1
votes
2 answers

How do you find a minimal polynomial?

I'm new to this subject and not brilliant at it at all! All of the tutorials I've found online have just been a bit too overhwelming and I have an exam tomorrow so is there any way somebody could explain as simply as they possibly could! An example…
Lindsay
  • 13
-1
votes
2 answers

Minimal polynom for $e^{2\pi i/3} + 2^{1/3}$

I have $\alpha = \exp(\frac{2\pi i}{3}) + 2^{\frac13}$ and I need to find a minimal polynomial for It in $\mathbb Q$, the "obvious" Path is to get the third Power of each term, but I'm struggling with It It would also be sufficient for me to show…
Daniel Moraes
  • 789
-2
votes
1 answer

Find the characteristic polynomial and minimal polynomial of the Transformation

Let $V=\{\,f(x) ∈ \Bbb R[x] : \deg f(x) ≤ n\,\}$ and T be the linear endomorphism on $V$ given by $$T(f(x)) = f(x + 1) + f(x − 1).$$ Find the minimal polynomial and the characteristic polynomial of the transformation $T$. So essentially what I am…
Bugcatcher123
  • 111
-4
votes
1 answer

rank of 4x4 matrix f(A) is

Let f(x) be the minimal polynomial of the 4x4 matrix A equal to 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 Then rank of the 4x4 matrix f(A)…
Nirmal singh
  • 3
1
2