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Abstract. Fix a Galois extension E/F of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S E denote the primes of E lying above those in S, and let O S E denote the ring of S E -integers of E. We then compare the Fitting ideal of especially for biquadratic extensions, where we compute the index of the higher Stickelberger ideal. We find a sufficient condition for the Fitting ideal to contain the higher Stickelberger ideal in the case where E is a biquadratic extension of F containing the first layer of the cyclotomic Z 2 -extension of F , and describe a class of biquadratic extensions of F = Q that satisfy this condition.

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