If you can program you can get a job. I think in your case you probably need to show you can do something by putting it up on the web or Github. PhDs from certain universities will probably get an automatic job offer regardless.
The question is do you want a job in your field of expertise? I am not familiar enough with companies looking at practical applications of abstract algebra but it's probably not as abundant as companies looking for people with a focus on statistics. I don't think this will hinder you much though as the industry is looking for smart people despite whatever field you're in.
Maybe this sounds stupid because it may be obvious but I think a big advantage is if you're able to both read math papers and translate them into actual working algorithms. It doesn't seem like there are many who know how to do both.