On the class number of the maximal real subfields of a family of cyclotomic fields

For any square-free positive integer m ≡ 10 (mod 16) with m ⩾ 26, we prove that the class number of the real cyclotomic field ℚ(ζ4m +ζ4m−1) is greater than 1, where ζ4m is a primitive 4mth root of unity.