Why is it "argument" instead of "arguement"?

It wasn't formed within English at all. According to the OED, this is the etymology of the word:

French argument (13th cent.), < Latin argūmentum , < arguĕre (or refashioning, after this, of Old French arguement , < arguer )

I don't know Latin, but I think I found a pattern:

• indument (obsolete) from "Latin indumentum garment, clothing, < induĕre"
• integument from "Latin integumentum covering, < integĕre to cover"
• involument (obsolete) from "late Latin involūmentum (Vulgate), wrapper, < involvĕre to involve"

I think Latin just drops the ending -ere from its verbs when adding -mentum.

As Laurel’s answer suggests, “dropping” the e comes from Latin, rather than English.

To give a little Latin background, arguere (“to enlighten,” “to plea,” “to make a case”) is the infinitive form of arguo (“I [do those things]”). The principle parts of a Latin verb are usually given as

• the 1st-person–present–indicative–active (arguo, “I make a case”),

• the infinitive (arguere, “to make a case”),

• the 1st-person–perfect–indicative–active (argui, “I made a case”), and

• the supine (argutum, also “to make a case,” but this time as a noun phrase¹).

So here you can see that -ere is the suffix used to form the infinitive form of the verb, which is perhaps the most “neutral” form it has (not being attached to any particular actor or tense). Notably, the imperative form of the verb takes the infinitive and drops the -re, so ordering someone to make a case would literally be argue (pronounced differently, though, since the e would not be silent and there would be no y sound, so closer to ar-goo-eh though that makes the e sound seem too strong).

Meanwhile, argumentum is also a Latin word, formed as arguo + mentum (“a tool or aid”), in short, a tool or aid for making your case. Argumentum doesn’t “drop” an e, it simply never had one. It instead drops o, as that is the 1st-person–present–indicative–active suffix and not really relevant to the noun that was made from it.

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1. _{For the curious, the difference between the infinitive and supine—which actually do exist in English as concepts, even though they use the same words—is roughly this: “I go to make a case,” is the infinitive, i.e. Arguere eo. “To make a case is the best choice,” is the supine, i.e. Argutum est optio optima.}

In UK the spelling includes the E.
In US it doesn't.
Easy. Though in Australia, once all spelling words the UK way, now adopting more American spelling in dropping the E in eg:
Arguement -> Argument.
Abridgement -> Abridgment. Judgement -> Judgment.
Colour -> Color.
All right -> Alright

Just these off top of my head. But I can't spell WITHOUT the E... just feels wrong LOL

The e isn't dropped in "argument", it's added in "argue". The root is "argu". The noun "argument" is formed by adding "ment" to the root. The present participle is formed by adding "ing" to the root. The present tense is formed by adding "e" to the root.