Question

Question:Let E be the field of algebraic numbers. Prove that E is an infinite dimensional algebraic extension of Q.

Short answer

Eis infinite dimensional algebraic extension of Q.

Step-by-step solution

Definition of irreducible polynomial.

Irreducible polynomial is a non constant polynomial that cannot be factored into the product of two non constant polynomial.

Showing that E is infinite dimensional algebraic extension.

Let Ebe any field of all algebraic numbers. That is $\left[E:Q\right]=\infty$.

Let p be any prime number and $n\ge 1$an integer. Then by Einstein irreducible criterion the polynomial $F\left(x\right)=x^n-p\in Q$is irreducible over Q.

Hence F(x) is minimal polynomial of $\sqrt[n]{p}$over Q.

$\sqrt[n]{p}\in E$and$\left[Q\left(\sqrt[n]{p}\right):Q\right]=n$

Therefore

$\left[E:Q\right]\ge n$

Since was arbitrary. Therefore $\left[E:Q\right]=\infty$.