# Answer to Question #12708 in Abstract Algebra for Hym@n B@ss

Question #12708

Prove that all ideals in field are trivial.

Expert's answer

Proof.

Let F be a field and J be a non-zero ideal of F, i.e. J is not

{0}.

We should prove that J=F.

Let u be a non-zero element of J, and

w = u^{-1}

be its inverse, so

wu = 1.

Then

1 = wu

belongs to J,

wehnce for any a from F we have that

a * 1 = a belogs to

J.

That is J=F, and so J is a trivial ideal of R.

Let F be a field and J be a non-zero ideal of F, i.e. J is not

{0}.

We should prove that J=F.

Let u be a non-zero element of J, and

w = u^{-1}

be its inverse, so

wu = 1.

Then

1 = wu

belongs to J,

wehnce for any a from F we have that

a * 1 = a belogs to

J.

That is J=F, and so J is a trivial ideal of R.

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