← 2014 Paper 2
UPSC 2014 Maths Optional Paper 2 Q4a — Step-by-Step Solution 15 marks · Section A
Rings: Definition, Axioms, Examples · Algebra · Read the full method →
Question
Prove that the set Q ( 5 ) = { a + b 5 : a , b ∈ Q } \mathbb Q(\sqrt 5)=\{a+b\sqrt 5:a,b\in\mathbb Q\} Q ( 5 ) = { a + b 5 : a , b ∈ Q }
Technique
Standard ring-axiom verification; closure under + + + × \times ×
Solution
Strategy. Verify each ring axiom in turn. Closure under + + + × \times × R \mathbb R R
Step 1 — Closure under addition
For x = a 1 + b 1 5 , y = a 2 + b 2 5 ∈ Q ( 5 ) x=a_1+b_1\sqrt 5,\;y=a_2+b_2\sqrt 5\in\mathbb Q(\sqrt 5) x = a 1 + b 1 5 , y = a 2 + b 2 5 ∈ Q ( 5 )
x + y = ( a 1 + a 2 ) + ( b 1 + b 2 ) 5 . x+y=(a_1+a_2)+(b_1+b_2)\sqrt 5. x + y = ( a 1 + a 2 ) + ( b 1 + b 2 ) 5 . Since a 1 + a 2 , b 1 + b 2 ∈ Q a_1+a_2,\;b_1+b_2\in\mathbb Q a 1 + a 2 , b 1 + b 2 ∈ Q Q ( 5 ) \mathbb Q(\sqrt 5) Q ( 5 )
Step 2 — Closure under multiplication
x y = ( a 1 + b 1 5 ) ( a 2 + b 2 5 ) = a 1 a 2 + a 1 b 2 5 + b 1 a 2 5 + 5 b 1 b 2 = ( a 1 a 2 + 5 b 1 b 2 ) + ( a 1 b 2 + a 2 b 1 ) 5 . xy=(a_1+b_1\sqrt 5)(a_2+b_2\sqrt 5)=a_1 a_2+a_1 b_2\sqrt 5+b_1 a_2\sqrt 5+5b_1 b_2=(a_1 a_2+5b_1 b_2)+(a_1 b_2+a_2 b_1)\sqrt 5. x y = ( a 1 + b 1 5 ) ( a 2 + b 2 5 ) = a 1 a 2 + a 1 b 2 5 + b 1 a 2 5 + 5 b 1 b 2 = ( a 1 a 2 + 5 b 1 b 2 ) + ( a 1 b 2 + a 2 b 1 ) 5 . Both coefficients are rational. So x y ∈ Q ( 5 ) xy\in\mathbb Q(\sqrt 5) x y ∈ Q ( 5 )
Step 3 — Additive identity and inverses
0 = 0 + 0 5 ∈ Q ( 5 ) 0=0+0\sqrt 5\in\mathbb Q(\sqrt 5) 0 = 0 + 0 5 ∈ Q ( 5 ) x = a + b 5 x=a+b\sqrt 5 x = a + b 5 − x = − a + ( − b ) 5 ∈ Q ( 5 ) -x=-a+(-b)\sqrt 5\in\mathbb Q(\sqrt 5) − x = − a + ( − b ) 5 ∈ Q ( 5 )
So ( Q ( 5 ) , + ) (\mathbb Q(\sqrt 5),+) ( Q ( 5 ) , + )
Step 4 — Multiplicative identity
1 = 1 + 0 5 ∈ Q ( 5 ) 1=1+0\sqrt 5\in\mathbb Q(\sqrt 5) 1 = 1 + 0 5 ∈ Q ( 5 ) x ∈ Q ( 5 ) x\in\mathbb Q(\sqrt 5) x ∈ Q ( 5 ) 1 ⋅ x = x ⋅ 1 = x 1\cdot x=x\cdot 1=x 1 ⋅ x = x ⋅ 1 = x
Step 5 — Other axioms
Commutativity of + + + x + y = y + x x+y=y+x x + y = y + x R \mathbb R R Associativity of + + + ✓.Commutativity of × \times × x y = y x xy=yx x y = y x x , y x,y x , y Associativity of × \times × ✓.Distributivity: x ( y + z ) = x y + x z x(y+z)=xy+xz x ( y + z ) = x y + x z ( y + z ) x = y x + z x (y+z)x=yx+zx ( y + z ) x = y x + z x
All ring axioms hold, plus × \times × Q ( 5 ) \mathbb Q(\sqrt 5) Q ( 5 ) commutative ring with identity (a commutative unital ring).
Answer
Q ( 5 ) is a commutative ring with identity. \boxed{\;\mathbb Q(\sqrt 5)\text{ is a commutative ring with identity.}\;} Q ( 5 ) is a commutative ring with identity.