Field extension and multiplicativity formula for degrees

Claim
Let L/K be a field extension, a ∈ L with odd degree [K(a) : K].
Then holds: K(a) = K(a2).

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

Proof

Part 1:
L is a field, we have: a ∈ L ⇒ a2 ∈ L

Furthermore: a2 ∈ K(a)

So we have a tower: K ⊆ K(a2) ⊆ K(a)

Now we can use the multiplicativity formula for degrees:
[K(a) : K] = [K(a) : K(a2)] [K(a2) : K]

Consider the polynomial p(x) = x2 -a2 in K(a2)

It has the root a.

The degree of the minimal polynomial of a over K(a2) is no greater than 2.

That means: [K(a) : K(a2)] ≤ 2

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

That implies: [K(a) : K(a2)] = 1 ⇔ K(a) = K(a2).

Part 2
Let be K = ℚ and a = √2. Then holds:
K(a2) = ℚ(2) = ℚ ≠ ℚ(√2) = K(a).