Yes, this is gluten free
None of these plants makes gluten: rice, corn, quinoa, amaranth, millet, sorghum and wild rice are unrelated to wheat, and the risk enters at the mill rather than in the field.
amaranth is covered by CiCi's entry on naturally gluten-free grains, which share this answer and the reasoning behind it.
Every one of these is the same ingredient. The number of names is a large part of why gluten slips past people who read every label.
Point CiCi at the barcode on the product you're holding and get this same reasoning in about two seconds, plus the carbs and the full nutrition panel. 360,000+ packaged foods are already read and waiting.
This answer comes from CiCi's ingredient lexicon — the same rules the scanner uses, not a separate article. Where reputable sources disagree, CiCi takes the conservative position and says so.