Question: May one learn during chazaras hashatz?
Answer: The Shulchan Aruch (OC 124:1) writes that the chazzan repeats the amida aloud so that everyone can fulfil their obligation of tefilla, even if they are unable to daven themselves. Although most people in shuls today are capable of davening themselves, the Kaf Hachaim (OC 124:2) writes that chazaras hashatz is more important than the quiet shemonei esrei.
The Mishna Berura (124:17) writes that one must not recite other tefillos or learn during chazaras hashatz even when it does not prevent them from answering amen, as others may be misled into thinking that it is even okay to talk. Elsewhere (125:1), he writes that it is fine to learn by thinking (hirhur) except during kedusha. R’ Moshe Feinstein (Igros Moshe OC 4:19) notes that one who isn’t listening properly to the chazaras hashatz is not considered to be part of the minyan at that time.
The Magen Avraham (124:8), Shulchan Aruch Harav (OC 124:6) and Baer Heitev (OC 124:4) quote opinions both for and against preventing those who learn during chazaras hashatz, even if they are answering amen to the chazan. The Kaf Hachaim (OC 124:16) sides with those who oppose it, adding that one should not even think about Torah at this time (See Rivevos Ephrain 5:61).
In conclusion, one must not learn Torah during chazaras hashatz.