Problem 10 (10 points): Let L/Q be a Galois extension over Q with Gal(L/Q) = A4. 1. How many subfields does L/Q contain with degree 1, 2, 3, 4, 6, 12 over Q respectively? 2. Prove that for each subfield K with degree 6 and each subfield F with degree 3, F is always a subfield of K. 3. Let K and F be fields in the last question. Is K/F a Galois extension? Is K/Q a Galois extension? Problem 10 (10 points): Let L/Q be a Galois extension over Q with Gal(L/Q) = A4. 1. How many subfields does L/Q contain with degree 1, 2, 3, 4, 6, 12 over Q respectively? 2. Prove that for each subfield K with degree 6 and each subfield F with degree 3, F is always a subfield of K. 3. Let K and F be fields in the last question. Is K/F a Galois extension? Is K/Q a Galois extension