Question

$\mathit{F}\mathbf{⊑}\mathit{D}\mathbf{⊑}\mathit{K}$Question:Let D be a ring such that if K is algebraic over F prove that D is a field.

Short answer

Dis a field.

Step-by-step solution

Definition of ring.

A field is a commutative ring with unity which every nonzero element has a multicative inverse.

Showing that D is a field.

Let Dbe a ring such that $F\subseteq D\subseteq K$and K is algebraic over F and K over F is algebraic extension.

It is enough to show every nonzero element has an inverse.

Let $u\in D$be any arbitrary element. Then$u\in D$implies $u\in D$. Therefore $D\subseteq K$.

As k is algebraic extension over F. So,u is algebraic over F.

This implies F(u) is algebraic extension of F.

Therefore$F\subseteq D,u\in D$

Since F(u) is a field. So,$u^{-1}\in F\left(u\right)\subseteq D$

This implies$u^{-1}\subseteq D$

As $u\in D$was arbitrary.

Therefore every element of Dhas inverse.