Question

Prove sqrt(2)+(3)/(sqrt(2)) is irrational​

Answer 1

Answer:

Yes.

If 2–√+3–√ were rational, then its square, 5+26–√ , would be rational. Consequently, 6–√ would be rational. Yet 6–√=ab with a,b relatively prime implies that a2=6b2 ; and basic number theory leads us to a contradiction [for consultation see the classical proof of the irrationality of 2–√ ].

Alternatively, the Galois automorphism a+b2–√↦a−b2–√ of Q(2–√) extends to a Galois automorphism of Q(2–√,3–√) [because the extension field is normal], and such an automorphism must send 3–√↦±3–√ . As such, it maps 2–√+3–√↦−2–√±3–√ , which is certainly not equal to 2–√+3–√ ; therefore 2–√+3–√ is irrational (because rational numbers are fixed by every automorphism).

Step-by-step explanation:

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