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1.
a. show that every subfield of complex numbers contains rational numbers
b. show that the prime field of real numbers is rational numbers
c. show that the prime field of complex numbers is rational numbers
2.
a. Let R be a domain. Prove that the polynomial f(x) is a unit in R[x] if and only if f(x) is a nonzero constant which is a unit in R.
b. Show that ([2]x + [1])^2 = [1] in (integers modulo 4)[x] Conclude that the statement in part (a) may be false for the commutative rings that are not domains. [ An element z element of R is called a nilpoint if z^m = 0 for some integer m greater than or equal to one. For any commutative ring R, it can be proved that a polynomial f(x) = a(sub 0) + a(sub 1)x +...+a(sub n)x^n element R[x] is a unit in R[x] if and only if a(sub 0) is a unit in R and a(sub i) is nilpoint for all I greater than or equal to 1.]
Timmy enjoys playing old-fashioned video games. He found 3 game systems on eBay and wants to purchase a Nintendo game system, an Atari game system, and a Sega arcade game system.
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If this problem is too difficult, then perhaps you may solve an easier one, just to give me some direction please. The only example I have is the Riemann function for the telegraph equation
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