Find th eminimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb{Q}$

rocedwrp
2020-11-10
Answered

Find th eminimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb{Q}$

You can still ask an expert for help

Jayden-James Duffy

Answered 2020-11-11
Author has **91** answers

To find the smallest degree polynomial f(x) with coefficients in Q, such that $f(\sqrt{2}+\sqrt{3})=0.$

The impo

rtant point is that the coefficients should all be in Q. For example,$x-(\sqrt{2}+\sqrt{3})$ is not allowed, as its coefficients are not rational numbers.

Now,$\sqrt{3}\notin \mathbb{Q}\sqrt{2}.$

So, we expect the minimal polynomial of$\sqrt{2}+\sqrt{3}$ to have degree $4({,}^{\prime}\sqrt{2}+\sqrt{3}\in \mathbb{Q}\left(\sqrt{2}\right)\cdot \left(\sqrt{3}\right)$ , a quadrantic extension of a quadrantic extension of $\mathbb{Q}$ .

We proceed to find this minimal polynomial by repeated squaring (to clear radicals and obtain rational coefficients)

$x=\sqrt{2}+\sqrt{3}\Rightarrow x-\sqrt{2}=\sqrt{3},$ squaring

${(x-\sqrt{2})}^{2}=3\Rightarrow {x}^{2}-2\sqrt{2}x+2=3$ , rearrange

${x}^{2}-1=2\sqrt{2}x$ , square again

$({x}^{2}-1)}^{2}=8{x}^{2}\Rightarrow {x}^{4}-2{x}^{2}+1=8{x}^{2$ , rearrange

Minimal polynomial is${x}^{4}-10{x}^{2}+1$

The impo

rtant point is that the coefficients should all be in Q. For example,

Now,

So, we expect the minimal polynomial of

We proceed to find this minimal polynomial by repeated squaring (to clear radicals and obtain rational coefficients)

Minimal polynomial is

asked 2020-11-20

Prove that in any group, an element and its inverse have the same order.

asked 2022-06-06

I have a problem understanding getting the KERNEL and IMAGE of a linear transformation. We have the following transformation given:

${\mathbb{R}}_{2}[x]\to {\mathbb{R}}_{2}[x]$

$(\varphi (p))(x)=(xp(x+1){)}^{\prime}-2p(x)$

We first have to find its matrix in basis

$\{1,x,{x}^{2}\}$

which I know how to get. The transformation matrix result is:

$\left[\begin{array}{ccc}-1& 1& 1\\ 0& 0& 4\\ 0& 0& 1\end{array}\right]$

How do I get the KERNEL and the IMAGE from it ?

${\mathbb{R}}_{2}[x]\to {\mathbb{R}}_{2}[x]$

$(\varphi (p))(x)=(xp(x+1){)}^{\prime}-2p(x)$

We first have to find its matrix in basis

$\{1,x,{x}^{2}\}$

which I know how to get. The transformation matrix result is:

$\left[\begin{array}{ccc}-1& 1& 1\\ 0& 0& 4\\ 0& 0& 1\end{array}\right]$

How do I get the KERNEL and the IMAGE from it ?

asked 2022-01-20

Two welders worked a total of 47 h on a project. One welder made $37/h, while the other made $39/h. If the gross earnings of the two welders was $1781 for the job, how many hours did each welder work? Using row- echron matrix.

asked 2022-04-06

For any vectors u, v and w, show that the vectors u+v, u+w and v+w form a linearly dependent set.

asked 2022-01-05

Show that the xy plane $W=(x,y,0)$ in $\mathbb{R}}^{3$ is generated by

(i)$u=\left[\begin{array}{c}1\\ 2\\ 0\end{array}\right]$ and $v=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$ (ii) $u=\left[\begin{array}{c}2\\ -1\\ 0\end{array}\right]$ and $v=\left[\begin{array}{c}1\\ 3\\ 0\end{array}\right]$

(i)

asked 2020-11-01

The equivalent polar coordinates for the given rectangular coordinates.

A rectangular coordinate is given as (0, -3).

A rectangular coordinate is given as (0, -3).

asked 2022-03-30

If u, v, w ∈ R n , then span(u, v + w) = span(u + v, w)