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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 2vectors (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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 Winter '09
 IgorDolgachev
 Linear Algebra, Algebra, Complex Numbers

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