Two students named Field are in the rolls of the Mordecai school, both from Mecklenburg, Virginia, both among the first students when the school opened in 1809. Eliza Field and Harriet Field were both at the school for six sessions, on and off between 1809 and 1813 for Harriet, 1814 for Eliza.
These might be Harriet and Eliza Field, the daughters of judge Hume Riggs Field (1772-1831) and his first wife Millicine (or Mildred) Young Field (1782-1827), of Mecklenburg Virginia. The dates might not work out, though: Harriet H. Field was born in 1800--so she was nine the year the school opened, that makes sense. Eliza, however, was younger--born in 1806--maybe too much younger to be a Mordecai student in 1809.
If Harriet's sister Eliza was the Eliza Field at the Mordecai school in 1809, she would have been three years old, and the youngest known student. So I suspect she was the Eliza who attended after 1812; but a different Eliza Field, maybe a cousin, might have been there in 1809 and early 1810. The name isn't so unusual, anyway. Assuming the daughters of Hume and Millicine Field attended the Mordecai school...
Harriet H. Field (1800-1850) married Charles Perkins in about 1821, and they had at least one child, Marietta. She was widowed around 1828, and died in Tennessee in 1850, aged 50 years.
Eliza Mildred Field (1806-?) married Charles Perkins' brother, Constantine Perkins (1792-1836) in about 1824, and they had at least three children, Constantine, Ann Eliza, and Virginia in Tuscaloosa, Alabama. Her husband served in the Alabama legislature, and as the state's attorney general.