Let K := {s + t√2:s,t ∈ Q}. Show that K satisfies the following: If x1,x2 ∈ K, then x1+x2 ∈ K and x1x2 ∈ K.?
Let K := {s + t√2:s,t ∈ Q}. Show that K satisfies the following: If x1,x2 ∈ K, then x1+x2 ∈ K and x1x2 ∈ K
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Let x1,x2 be in K. Then x1=s1+t1sqrt(2) for some
s1,t1 in Q and similarily for x2.
Observe that
x1+x2=(s1+s1)+(t1+t2)sqrt(2) which is in K.
Also
x1*x2=[s1+t1sqrt(2)]*[s2+t2sqrt(2)]
=s1*s1+s1*t2sqrt(2)+s2*t1sqrt(2)+2*t2*t2
=(s1*s1+2t1*t2)+(s1*t2+s2*t1)*sqrt(2).
Since s1*s1,s2*s2,s1*t2,s2*t1 are in Q
since Q is closed under multiplication and addition,
it follws that x1*x2 is in Q.
Hope that helps.
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