Does brushing your teeth break the fast?

No, as long as you do not swallow water or toothpaste.

Does an injection break the fast?

Non-nutritive injections (vaccines, insulin) do not break it in the view of many scholars. Nutritive IV drips do.

Your fast is still valid. Stop as soon as you remember and complete the day.

Learn

Fasting means refraining from food, drink and marital relations from the start of Fajr until sunset, with the intention to fast for Allah.

Eating or drinking on purpose, including smoking.
Sexual intercourse during the day; this also requires kaffarah (expiation) in most schools.
Deliberate vomiting.
Menstruation or post-natal bleeding, even moments before sunset.
Anything taken into the stomach through the mouth or nose, such as nasal drops reaching the throat, in the majority view.

A broken fast is made up later with one day (qada).

Eating or drinking by mistake: "Whoever forgets and eats or drinks, let him complete his fast, for Allah has fed him" (Bukhari, Muslim).
Swallowing your own saliva, tasting nothing.
Brushing teeth without swallowing water or toothpaste (many prefer a miswak or brushing before Fajr).
Unintentional vomiting.
Showering, swimming without swallowing water, wearing perfume.
Blood tests and, in the view of many contemporary scholars, non-nutritive injections. Ask a scholar about nutritive drips.

The sick and travellers: they make up the days later (2:184).
Pregnant and breastfeeding women who fear for themselves or the baby: they make up the days (some schools add fidyah).
The elderly and the chronically ill who cannot fast: they give fidyah, feeding one poor person for each day.