Thread: Irreducibilty in F[x]

Let f:[0,1]→R be defined by f(x)=x, show that the polynomial x^2+f in T[x] is irreducible, where T is the ring of continuous real valued function on [0,1].

For irreduciblity do i just show that this does not have any roots. by evaluating the equation x^2 + f, at 0 and 1?

Originally Posted by empressA88

Let f:[0,1]→R be defined by f(x)=x, show that the polynomial x^2+f in T[x] is irreducible, where T is the ring of continuous real valued function on [0,1].

For irreduciblity do i just show that this does not have any roots. by evaluating the equation x^2 + f, at 0 and 1?

the question is kind of strange because the continiuity condition is useless here. you need to prove that no real-valued (continuous) function on the interval [0,1] exists such that

for all which is trivially true.

for all

so once i prove this equation, it should cover it?

Originally Posted by empressA88

for all

so once i prove this equation, it should cover it?

Of course: since the polynomial has degree , it reducible iff it has a root in s.t. , since the zero element in the ring is the zero map...and this is what NCA told you is trivially true.

Tonio

Originally Posted by empressA88

for all

so once i prove this equation, it should cover it?

yes, and there's nothing to prove! just put x = 1, or any positive number, to see that the equality cannot hold.

ok..thank you