Paul est fou.

Est is the third-person singular form of the verb être (to be).

The subject is Paul, which is like he in English, so French uses il est / Paul est.

A quick comparison:

• je suis = I am
• tu es = you are
• il / elle est = he / she is

So:

• Paul est fou = Paul is crazy

Because fou is an adjective, not a noun, in this sentence.

In French, just like in English, after to be / être, you usually use an adjective directly:

• Paul est fou = Paul is crazy
• Marie est heureuse = Marie is happy

You would not say Paul est un fou unless you specifically mean Paul is a madman / a crazy person, where fou is being used as a noun.

So:

• Paul est fou = adjective
• Paul est un fou = noun, different nuance

Because fou is describing Paul after the verb être.

This is normal French sentence structure:

• subject + verb + adjective

So:

• Paul = subject
• est = verb
• fou = adjective describing Paul

This works very much like English:

• Paul is crazy

Yes. French adjectives usually agree with the person or thing they describe in gender and number.

For fou, the forms are:

• masculine singular: fou
• feminine singular: folle
• masculine plural: fous
• feminine plural: folles

Examples:

• Paul est fou.
• Marie est folle.
• Paul et Jean sont fous.
• Marie et Anne sont folles.

So in your sentence, fou is used because Paul is masculine singular.

This is just an irregular adjective pattern that you have to learn.

Some French adjectives change more noticeably in the feminine form. With fou, the feminine is:

• fou → folle

This kind of change also appears in a few other adjectives, for example:

• mou → molle
• vieux → vieille

So folle is not something you can guess from a simple rule like add -e; it is a form to memorize.

A common pronunciation is:

/pɔl ɛ fu/

Roughly:

• Paul sounds like Pol
• est sounds like eh
• fou sounds like foo

A few useful notes:

• In Paul, the l is pronounced.
• In est, the final t is not pronounced here.
• ou in fou sounds like oo in food.

So the whole sentence sounds roughly like:

Pol eh foo

No, not between est and fou.

Liaison happens when a normally silent final consonant is pronounced before a following word that begins with a vowel sound. But fou begins with f, a consonant sound, so there is no liaison there.

Also, between Paul and est, there is no special liaison sound added either. You simply say:

Paul est fou → /pɔl ɛ fu/

It can be.

Most basically, fou means crazy / mad, and depending on context it can sound:

• literal
• insulting
• joking
• exaggerated

For example, it might describe someone as mentally unstable, but it can also be used more loosely, like saying someone is acting wild or irrational.

So tone matters a lot. In isolation, Paul est fou can sound fairly strong.

Yes, but usually with extra words.

For example:

• Paul est fou de musique. = Paul is crazy about music.
• Elle est folle de lui. = She is crazy about him.

In your sentence, Paul est fou, with no complement after it, the meaning is simply that Paul is crazy / mad.

Because Paul is a proper name.

In French, proper names usually do not take an article:

• Paul est fou
• Marie arrive
• Jean parle

That is the same basic idea as in English: we normally say Paul, not the Paul.

Yes, but it does not mean exactly the same thing.

• Paul est fou = Paul is crazy
  
  • fou is an adjective
• Paul est un fou = Paul is a madman / Paul is a fool / Paul is one crazy guy
  
  • fou is a noun

So both are possible, but they are different structures and have different nuances.

Yes. It is a very standard French sentence.

The pattern is:

• Paul = subject
• est = verb
• fou = adjective

So it follows a normal structure that matches English quite closely:

• Paul is crazy

This makes it a very useful beginner sentence for learning how être works with adjectives.