# Central simple algebra of dimension 4

Suppose $A$ is a $F$-central simple algebra with maximal subfield $E$ such that $[A:F] = 4$. if $N_{E/F}(E^*) \ne F^*$, then $A$ is a division algebra.

Is this even true? If it is true how i can prove it? I know we have $A \simeq M_2(F)$ or $A$ is a division ring so we must show that $A$ isn't isomorphic with $M_2(F)$.

• No, it's not true. All quadratic extensions $E/F$ are contained in $M_2(F)$. – Kimball Jul 23 '18 at 0:42
• For example consider $\mathbb{C}/\mathbb{R}$ and set: $$A = \left\lbrace \begin{pmatrix} x & y\\\overline{y} & \overline{x} \end{pmatrix} :x,y\in \mathbb{C}\right\rbrace \subseteq M_2(\mathbb{C})$$. so A is a $\mathbb{R}$-central simple algebra and isn't division ring. is this counter-example correct? – s.Bahari Jul 23 '18 at 16:18
• Yes, your $A$ is isomorphic to $M_2(\mathbb R)$. – Kimball Jul 24 '18 at 1:33
• Thanks for answer. – s.Bahari Jul 24 '18 at 15:09