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!
