It is currently Thu Jun 20, 2019 4:24 am

 All times are UTC [ DST ]

 Page 1 of 1 [ 3 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: $\mathbb{R}^5$ over $\mathbb{R}$Posted: Mon Jan 18, 2016 3:52 am
 Team Member

Joined: Mon Nov 09, 2015 1:36 am
Posts: 460
Location: Ioannina, Greece
It is possible to define a multiplication in $\mathbb{R}^5$ such that, with the usual addition and multiplication between elements of $\mathbb{R}$, the $\mathbb{R}^5$ becomes a field which contains the $\mathbb{R}$?

NOTE: I don't have a solution for this.

_________________
Grigorios Kostakos

Top

 Post subject: Re: $\mathbb{R}^5$ over $\mathbb{R}$Posted: Mon Jan 18, 2016 3:53 am
 Team Member

Joined: Tue Nov 10, 2015 8:25 pm
Posts: 314
Suppose that a multiplication can be defined in $\displaystyle \mathbb{R}^5$ so that it becomes a field extension of $\displaystyle \mathbb{R}.$ Then $\displaystyle \mathbb{R}^5$ can be viewed as a vector space over $\displaystyle \mathbb{R}$ and obviously $\displaystyle \dim_{\mathbb{R}}\left( \mathbb{R}^5 \right) = 5.$ Therefore the degree $\displaystyle \left[ \mathbb{R}^5 : \mathbb{R} \right ]$ of the field extension $\displaystyle \mathbb{R}^5 / \mathbb{R}$ equals 5. Let $\displaystyle \alpha \in \mathbb{R}^5 \smallsetminus \mathbb{R}.$ Then we have that
$\displaystyle 5 = \left[ \mathbb{R}^5 : \mathbb{R} \right ] = \left[ \mathbb{R}^5 : \mathbb{R}(\alpha) \right ] \left[ \mathbb{R}(\alpha) : \mathbb{R} \right ]$
which means that either
$\displaystyle \left[ \mathbb{R}^5 : \mathbb{R}(\alpha) \right ] = 5 \text{ and } \big[ \mathbb{R}(\alpha) : \mathbb{R} \big ] = 1 \; (*)$
or
$\displaystyle \left[ \mathbb{R}^5 : \mathbb{R}(\alpha) \right ] = 1 \text{ and } \big[ \mathbb{R}(\alpha) : \mathbb{R} \big ] = 5 \; (**)$

- If $\displaystyle (*)$ is true, then $\displaystyle \left[ \mathbb{R}(\alpha) : \mathbb{R} \right ] = 1$ means that $\displaystyle \mathbb{R}(\alpha) = \mathbb{R}$, which is a contradiction, since $\displaystyle \alpha \notin \mathbb{R}.$

- If $\displaystyle (**)$ is true, then $\displaystyle \left[ \mathbb{R}(\alpha) : \mathbb{R} \right ] = 5$ means that $\displaystyle irr\left( \alpha , \mathbb{R} \right)$ is of odd degree. However, this cannot be true, since it is known that every polynomial of odd degree over $\displaystyle \mathbb{R}$ has a real root and, thus, $\displaystyle \partial \left( irr\left( \alpha , \mathbb{R} \right) \right) \leq 4 .$

Hence the desired multiplication cannot be defined.

In fact, with a similar argument we can prove that there is no field extension of $\displaystyle \mathbb{R}$ of odd degree stricktly greater that 1, which means that $\displaystyle \mathbb{R}$ is the only field extension of $\displaystyle \mathbb{R}$ of odd degree!

Top

 Post subject: Re: $\mathbb{R}^5$ over $\mathbb{R}$Posted: Thu Sep 08, 2016 2:05 pm

Joined: Fri Aug 12, 2016 4:33 pm
Posts: 15
We have
1)Every finite extension is algebraic
2)The algebraic closure of real numbers are
the complex numbers
3)Every finite extension of real numbers are
the complex numbers.

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 3 posts ]

 All times are UTC [ DST ]

#### Mathimatikoi Online

Users browsing this forum: No registered users and 2 guests

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot post attachments in this forum

Search for:
 Jump to:  Select a forum ------------------ Algebra    Linear Algebra    Algebraic Structures    Homological Algebra Analysis    Real Analysis    Complex Analysis    Calculus    Multivariate Calculus    Functional Analysis    Measure and Integration Theory Geometry    Euclidean Geometry    Analytic Geometry    Projective Geometry, Solid Geometry    Differential Geometry Topology    General Topology    Algebraic Topology Category theory Algebraic Geometry Number theory Differential Equations    ODE    PDE Probability & Statistics Combinatorics General Mathematics Foundation Competitions Archives LaTeX    LaTeX & Mathjax    LaTeX code testings Meta