Finitely Generated Abelian Groups and Similarity of Matrices over a Field by Christopher Norman

Author:Christopher Norman
Language: eng
Format: epub, pdf
Publisher: Springer London, London

(b)Let E be a field with subfield F and let L be an intermediate subfield, that is, L is a subfield of E and F⊆L⊆E. Suppose that L is a finite extension of F with basis u 1,u 2,…,u m where m=[L:F] and suppose also that E is a finite extension of L with basis v 1,v 2,…,v n where n=[E:L]. Prove that the mn elements u i v j of E (1≤i≤m,1≤j≤n) form a basis of E over F. Deduce that E is a finite extension of F and [E:F]=[E:L][L:F].

(c)Let E be a finite field with subfield F of order q and let [E:F]=n.

(i) Show that every subfield L of E satisfies |L|=q d where d|n. Conversely show (q d −1)|(q n −1) and for each positive divisor d of n, and hence show that is the unique subfield of E with |L|=q d .

Hint: Consider the fixed field of θ d where θ is the Frobenius automorphism of E.

(ii) Let L and M be subfields of E with F⊆L∩M and so |L|=q d and |L|=q e . Deduce from (i) above that |L∩M|=q gcd{d,e}.

(iii) Let be the factorisation of n>1 into positive powers of distinct primes p 1,p 2,…,p k . For each subset X of {1,2,…,k} write π X =∏ j∈X p j , i.e. π X is the product of the primes p j for j∈X and π ∅=1. Let L j be the subfield of E with |L j |=q n /p j (1≤j≤k). Use induction on s=|X| and (ii) above to show |⋂ j∈X L j |=q n /π X . The sieve formula now asserts that the number of elements of E which are not in any subfield L of E with F⊆L≠E is r=∑ X (−1)|X| q n /π X , the summation being over the 2 k subsets X of {1,2,…,k}. Using Questions 3(c) and 7(a) above, explain why r/n (Dedekind’s formula) is the number of monic irreducible polynomials of degree n over .

Verify that there are 335 monic irreducible polynomials of degree 12 over ℤ2. Verify that there are 670 monic irreducible polynomials of degree 6 over . Calculate the numbers of monic irreducible polynomials of degree 12 over ℤ3 and of degree 6 over .

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