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**Additional info for Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue**

**Example text**

Proof. 20 is obviously true when < S >= {0}. Now let dim(< S >) = k ≥ 1 and suppose {v1 , . . , vk } is a basis of < S >. We need to show that dim(S ⊥ ) = dim(< S >⊥ ) = n − k. Note that x ∈ S ⊥ if and only if v1 · x = · · · = vk · x = 0, which is equivalent to saying that x satisfies AxT = 0, where A is the k × n matrix whose ith row is vi . The rows of A are linearly independent, so AxT = 0 is a linear system of k linearly independent equations in n variables. From linear algebra, it is known that such a system admits a solution space of dimension n − k.

4) to obtain α 2 + α 6 = (1 − α) + (1 − α)3 = 2 − α − α 3 = 2 − α − α(1 − α) = 2 − 2α + α 2 = 0. Hence, the minimal polynomial of α 2 is 1 + x 2 . In the same way, we may obtain the minimal polynomial 2 + 2x + x 2 of α 5 . The following result will be useful when we study cyclic codes in Chapter 7. 11 Let n be a positive integer with gcd(q, n) = 1. Suppose that m is a positive integer satisfying n|(q m − 1). Let α be a primitive element of Fq m and let M ( j) (x) be the minimal polynomial of α j with respect to Fq .

Ii) Determine all the subfields in (a) F212 , (b) F218 . 23 Factorize the following polynomials: (a) x 7 − 1 over F2 ; (b) x 15 − 1 over F2 ; 31 (c) x − 1 over F2 ; (d) x 8 − 1 over F3 ; (e) x 12 − 1 over F5 ; (f) x 24 − 1 over F7 . 24 Show that, for any given element c of Fq , there exist two elements a and b of Fq such that a 2 + b2 = c. 25 For a nonzero element b of F p , where p is a prime, prove that the trinomial x p − x − b is irreducible in F pn [x] if and only if n is not divisible by p.