In your case the function field of $X_{\eta}$ is the same as the function field of $X$. Thus the following general remark answers your question affirmatively.

Assume $Y$ is an integral scheme and $L$ is an algebraic extension of the function field $K(Y)$ of $Y$. Let $\pi\colon X \to Y$ be an integral morphism of schemes such that $X$ is integral and normal and such that $\pi$ induces on function fields the extension $K(Y) \subset L = K(X)$. Then $X$ is the normalization of $Y$ in $L$. In fact this follows essentially from the definition of "normalization" and the fact that integral ring homomorphisms are stable under localization.

Is this morphism the normalization of P^1 in  this curve
My car is a 2005 BMW 525i E60 with about 180,000km on the clock. It has been well looked after and serviced. I don't get dealer services, but go to a mechanic who has always been good and uses the same oils and genuine parts that BMW use. He used to be a BMW tech for about 25 years before going out on his own and is considerably cheaper.

Before we proceed, I apologise for the long story and hope you can bear with me. Make yourself a tea or coffee before you read this. I greatly appreciate any help anyone has to offer.

The story begins about 2 weeks ago when I took my car to the mechanic for a service. I was experiencing issues where the car was sluggy and I had an oil leak, simple enough I thought.

The mechanic called me and said they discovered a couple of inches of the fuel line beneath the car had been crushed (the hose is metal bent at weird angles and whatnot) and brake lines had also been squashed basically flat. The lines were squashed by a loose chunk of concrete coming off of a driveway. They called BMW and they were quoted $1300 AUD and a 3 week wait as the lines had to come from Germany.

They phoned around and were able to find second hand lines for about $400. They fitted the new lines and discovered the source of the leak was actually the oil filter, at first they thought the mount was cracked, but the car just needed a new oil filter. They also discovered a faulty part (a thermostat they had only replaced not even a year ago) still under warranty that they replaced. An engine mount near the alternator was also replaced, I recall they mentioned having to remove the alternator and whatnot to get to it.

After spending $2200 AUD on the repairs (a little more than I was expecting), I picked the car up and for a day it was perfect. It had a new-found sense of performance, was smoother than ever. Then the second day rolled around and my fiance calls me to tell me the car was blowing white smoke.

She picked me up and the car would pour out copious amounts of white smoke when taking off from lights, going up hills, etc. It wasn't a small amount either, it was literally clouds. It didn't have a sweet smell, more of a fume smell. The car was also idling rough and driving pretty rough as well, wasn't overly lacking power though.

I know what you are thinking and yes, it was my first thought too: the head gasket is blown and coolant and oil were mixing together. I had a sinking feeling in my stomach.

After the car had cooled down, I checked the oil and coolant. The oil was pretty much empty, it was below the minimum mark and the coolant was also a quarter down. Looking at the mechanic invoice it said they used about 6 litres of synthetic oil, where did it go I wondered. I checked the coolant and it was clean, I checked the oil and it was clean. I've seen what a blown head gasket looks like this wasn't looking the case. The oil had vanished, and some of the coolant as well, but it didn't appear as though the two were mixing.

Every time the car was started from a stationery position, it would blow lots of white smoke as it did when the car was being driven. Moisture getting into the engine somewhere seemed to be the most likely cause.

Because it was a late Friday evening and the mechanic workshop wasn't open until Monday, I decided to carefully fill the oil and coolant back up, I had some demineralised water and I also had some synthetic motor oil on hand. I filled the oil to just a little below the minimum mark and the coolant back to the full mark. I would call the mechanic on Monday and do some observational testing over the weekend.

As soon as I topped up both fluids, I started the car and no white smoke. I let it run for about 15 minutes. I turned it off, checked fluids and they were still full. The oil still appeared to be clean. I drove the car around the block and the white smoke was still gone. This was a good sign, but also a sign perhaps they forgot to top up the oil they charged me for, had it done damage I wondered?

The next morning I started the car and a puff of white smoke emerged and then cleared. Driving it however seemed to pose no problem, the white smoke would only appear after the car had been turned off and sitting for quite a while and first started. I checked the fluids and the oil was fine, the coolant had gone down a really tiny bit. Because I always park my car on a drip tray I noticed there was a water like oil substance on the tray, it wasn't a large amount, but enough to make me take notice. The oil had not gone down which made it all the more weird, where was the oil like colour and substance coming from?

Sunday rolled around the subsequently the same thing. No oil loss, coolant had gone down a bit more, white smoke on initial start after sitting and then cleared. I was hopeful when I took it to the mechanic on Monday it would be resolved.

I took it to the mechanic finally, I told them how the oil was empty and I topped it up, I also explained the coolant loss and they got me to start the car. Because the cloud of white smoke only happens on start, they couldn't see it. They said they could see a tiny bit of exhaust smoke and then subsequently tried telling me because of the age of the car (it's a 2005) that it was most likely the valve stem seals.

They then sold me a bottle of what looked like water which they poured into the oil for softening up the valve stem seals and said to keep an eye on it. They said if I notice any troubles, to bring it back. I stupidly went along with it even after explaining the oil loss and coolant loss. They offered no possible explanation as to why the oil was empty, I assumed it wasn't topped up. 

All of these issues are new ones I didn't have before I took the car to them, why would the valve stem seals all of a sudden cause me an issue after coincidentally getting it looked over and work done?

A couple of days had passed and the white smoke issue had temporarily disappeared and I thought, oh, maybe they were right. Then I noticed the smoke had returned again on start, I checked the fluids again and the oil was clean and fine. There were no signs of oil in the coolant or coolant in the oil. I however had noticed the coolant had dropped to about the half way mark now and the watery like oil was still leaking onto the drip tray.

I called them up and they said to bring it back down, I was determined to get them to actually look at the car this time. It was raining and I parked in the car park. The mechanic dude came out and I explained the situation. He offered no possible explanation for the coolant loss. It was also coincidentally raining and has been for the past few days. He said the water from the rain would have washed whatever was leaking from underneath and to bring it back when it's not raining.

What is going on here? What are my rights? I feel so helpless and trapped at the moment. I feel as though they're trying to avoid looking at the car properly maybe because they know they're responsible and scared maybe they blew the head gasket which is an expensive job.

What should I say to them next time I see them to make sure it gets fixed and looked at without incurring cost upon myself? I know it's hard, but what are some potential issues that could be responsible for my symptoms? Coolant loss, not going into the oil and oil not going down.
