513HW1 - Math. 513. Homework 1 In the following Q, R, C...

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Unformatted text preview: Math. 513. Homework 1 In the following Q, R, C denote the filed of rational, real and complex numbers. Also Fp denotes the Galois field of p elements. 1. Check whether the following set F forms a field with respect to given operations. √ (a) F = {a + b 2 ∈ R : a, b ∈ Q} considered as a a subset of R with respect to the usual operations of addition and multiplication. (b) F = {a + bi ∈ C : a, b, c ∈ Q} considered as a a subset of C with respect to the usual operations of addition and multiplication. √ √ (c) F = {a + b 2 + c 3 ∈ R : a, b, c ∈ Q} considered as a a subset of R with respect to the usual operations of addition and multiplication. (d) F is the set of matrices ab −b a , where a, b ∈ R with respect to matrix multiplication and addition. (e) Let F1 and F2 be two fields. Let F = F1 × F2 of pairs (a, b), a ∈ F1 , b ∈ F2 with operations (a, b) + (a , b ) = (a + a , b + b ) and (a, b) · (a , b ) = (aa , bb ). (f) ) F = R2 is the set of 2-vectors (a, b) with operations (a, b) + (a , b ) = (a + a , b + b ) and (a, b) · (a , b ) = (aa − bb , ab + a b). 2. Let F be a field. Deduce form the axioms that (−1)a = −a and 0 · a = 0 for any a ∈ F . 3. Let F be a set consisting of 4 elements {x, y, z, w}. Define operations + and · satisfying the axioms of a field. 4. What is the smallest subfield of R (i.e. a field contained in R but not containing any other filed)? 5. Solve the following system of linear equations over the given field F . x1 + x2 − x3 = 0 −x1 + 4x2 + x3 = 0 x2 + x3 = 0. (a) F = R (b) F = F5 ...
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