I've seen some other posts were already posted on this matter, but none of them seem to be oriented to performing the actual fourier transform of $\delta(x)$. I've just been introduced the Fourier transform and I'm having some trouble to check some of its properties, one of the most basic being:
$$\mathcal{F}(\delta) = 1$$
I keep getting a square root factor which I can't seem to get rid of:
$$\mathcal{F}(\delta) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-i\xi x}\delta(x)\cdot dx = \frac{1}{\sqrt{2\pi}} e^{0} = \frac{1}{\sqrt{2\pi}} \neq 1$$
Naturally, I get the same result when applying the integral definition of $\delta$:
$$\sqrt{2\pi} \delta(x) = \int_{-\infty}^\infty e^{ix\eta}\cdot d\eta$$
Introducing this expression inside the Fourier transform:
$$\mathcal{F}(\delta)\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-i\xi x}\left(\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{ix\eta}\cdot d\eta \right )dx$$
which leads me to the same conclusion as above.
My best guess is this result might be related to the normalisation factor $\frac{1}{\sqrt{2\pi}}$ and not to my understanding of Dirac's delta function, although this could also be true. Quite frankly, I'd like to expand my knowledge of these functions as they are ubiquitous in theoretical Physics, but my plain Physics degree doesn't cover them as much as I'd like. If you could recommend some resources to learn about these useful tools, I'd be more than grateful.