My girlfriend is having a baby thur artificial insemination and we have been together for three years and I want to adopt the baby can I legally adopt it we live in Alabama

Alabama's adoption laws are pretty open. But with artificial insemination there may not be laws to specifically cover it.

The likelihood is that you can adopt, but you may need to married first. I would consult a local attorney since this may require case law research. You will need an attorney to prepare the paperwork in any case.

Is this a same-sex relationship?

The laws are somewhat murkier with same-sex partners in Alabama, and you will definitely need to have a lawyer (even more so than usual) to proceed with an adoption in a same-sex relationship. ESPECIALLY since Alabama expressly forbids same-sex marriages at this time---so a step-parent adoption is not possible. And since you are not adopting together, it may be that she would have to give up her parental rights for you to be able to adopt.