She decided to undertake a PhD, which would take several years.

undertake 是动词，由前缀 under-（表示"在...之下"）和 take（承担、接受）组成，意思是"承担、着手做（某项任务或工作）"。在这里指"开始进行（攻读博士学位）这一长期学术任务"。