For this question, I am stuck on proving the associatitivty condition for the set to be a field. Here is what I have so far. Can anyone please help me?
Is the set $\Bbb Q[ √ 7] := {{a + b √ 7|a, b ∈ \Bbb Q}}$ with addition and multiplication defined by $(a + b √ 7) ⊕ (c + d √ 7) =(a + c) + (b + d) √ 7$ for all a, b, c, d ∈ Q, $(a + b √ 7) ⊗ (c + d √ 7) =(ac + 7bd) + (ad + bc) √ 7$ for all a, b, c, d ∈ Q, a field?
$[(a + b√7) ⊕ (c + d√7)]⊕(e+f√7)$
= $(a + b√7) ⊕ [(c + d√7)⊕(e+f√7)]$
= $(a + b√7) ⊕ [(c+e)+(d+f)√7]$
$[(a + b√7) ⊗ (c + d√7)]⊗(e+f√7)$
= $(a + b√7) ⊗[(c + d√7)⊗(e+f√7)]$
= $(a+b√7)⊗[(ce + 7df) + (cf + de)√7$