what polymer is synthesized during transcription?

The polymer synthesized during transcription is RNA, most specifically messenger RNA (mRNA), which is a polymer of ribonucleotides.
During transcription, an enzyme called RNA polymerase uses one strand of DNA as a template and links ribonucleotides together to form an RNA strand in the 5′→3′ direction. The resulting mRNA carries genetic information from DNA to ribosomes, where it can be translated into a polypeptide (protein).
In summary: transcription makes an RNA (usually mRNA) polymer, not DNA or protein.