Since $A$ is normal, for any minimal prime ideal $\p\in \spec A$ , $A/\p$ is integrally closed in its quotient field $\qf{A/\p}$ (which is a real closed field). And so, by [SV] Propositon 2, $A/\p$ is a real closed integral domain for every minimal prime ideal $\p\in \spec A$. By [Capco] Corollary 102 it follows that $A$ is then real closed.

Define $i:=\sqrt{-1_A}$. Then clearly $i\in T$, so by (classical) Artin-Schreier Theorem $\qf{A/\p}$ is a real closed field and $$\qf{A/\p}[i_\p] = \qf{T/\tilde\p}$$ where $i_\p = i\mod\tilde\p$.

- [Capco]
**Jose Capco**,*Real Closed * Rings*. PhD Dissertation, October 2010, Universität Passau, Germany. - [HochsterTIC]
**M. Hochster**,*Totally Integrally Closed Rings and Extremal Spaces*. Pacific Journal of Mathematics 1970, Vol. 32, No. 3, p.767-779. - [comalg]
**R.B. Ash**,*A Course in Commutative Algebra*. Accessed: 02.2005 - [SV]
**N. Sankaran, K. Varadarajan**,*Formally real rings and their real closures*. Acta Math. Hungarica 1996, vol. 19, p. 101-120