**Claim**

Let L/K be a field extension, a ∈ L with odd degree [K(a) : K].

Then holds: K(a) = K(a^{2}).

It is false for even [K(a) : K].

**Proof**

**Part 1:**

L is a field, we have: a ∈ L ⇒ a^{2} ∈ L

Furthermore: a^{2} ∈ K(a)

So we have a tower: K ⊆ K(a^{2}) ⊆ K(a)

Now we can use the multiplicativity formula for degrees:

[K(a) : K] = [K(a) : K(a^{2})] [K(a^{2}) : K]

Consider the polynomial p(x) = x^{2} -a^{2} in K(a^{2})

It has the root *a*.

The degree of the minimal polynomial of *a* over K(a^{2}) is no greater than 2.

That means: [K(a) : K(a^{2})] ≤ 2

If the degree was equal to 2, [K(a) : K] would be even, what is false by hypothesis.

That implies: [K(a) : K(a^{2})] = 1 ⇔ K(a) = K(a^{2}).

**Part 2**

Let be K = ℚ and a = √2. Then holds:

K(a^{2}) = ℚ(2) = ℚ ≠ ℚ(√2) = K(a).