I understand that in the sentence “ich möchte den blauen Regenschirm haben”: the adjective changes from blau to blauen as it is applied to a masculine object of the sentence. It takes the -en ending from den.
By that pattern, in the sentence “ich möchte das große Buch haben”, shouldn’t the adjective groß change to großes? Here too, Buch is a neutral object, hence the adjective ought to take on the -s ending from das right? Why is that incorrect?
das in the sentence ich möchte das große Buch haben is acting as a determiner of the gender of Buch. After the determiner is present, one must then conjugate the adjective endings as seen in this table:
This table and more information can be found on the University of Michigan’s website.
A nominal phrase or nominal group in German and some other languages consists of a single or no determiner ‘DET’ (i.e. article ‘ART’ or pronoun ‘PRON’), possibly multiple adjective ‘ADJ’ and a noun ‘SBST’ as its head. Some languages may show the grammatical agreement of these group members by affixing the same morpheme to all of them. German grammar does not work that way, so drop that wrong mental model now!
In German, usually the determiner of a group bears the clearest information about case, number and gender, fused into a single suffix. It always does if it is a definite article (der/das/die) or demonstrative pronoun (dies+); adjectives are inflected “weakly” then and can only end in -e or -n. For masculine and neuter nouns, the suffix shifts to the adjective(s) in Nom and NeutAcc if an indefinite article (ein+) or possessive pronoun (e.g. mein+) is used as determiner.
Only if the determiner is a numeral ‘NUM’ (i.e. a number or other quantifier), which is logically always plural, or is left out altogether, all markers move to the adjective(s), but the suffix changes from -s to -n for Gen Masc/Neut. They are said to inflect “strongly”. While numbers only decline in Gen, other quantifiers (viel+, einig+ …) usually inflect like the adjective.
Most nouns will exhibit case suffixes only for Gen (Masc/Neut) or Dat (Pl) or not at all (Fem), and sometimes these coincide with those of the attributes, e.g. des (schönen) Kindes and den schönen Kindern. Unlike attributes, nouns often have a special inflection base for the plural which can be formed by umlauting their stem or acquiring a suffix or both – any case suffix follows.
While nouns do not decline by gender (except for adjectives used as nouns), their final non-inflectional morpheme always determines it, e.g. +in → Fem. This effectively includes the plural morpheme, so German never treats gender and number separately, i.e. they’re one and the same category. However, verbs change by number only, as do 1st and 2nd personal pronouns, and interrogative pronouns depend on animate state in Nom and Acc using the same morphs as Masc vs. Neut elsewhere.
Also remember that only masculine attributes may have forms for all four cases, whereas neuter and plural ones don’t distinguish Acc from Nom and feminine ones furthermore don’t distinguish Dat from Gen.
What you are recognising as a pattern — the ending of the article being repeated in the adjective ending — is, in fact, not one.
German — like Sanskrit, English to a lesser extent and most other Indo-European languages — is a fusional language, meaning that an ending carries a lot of grammatical information in a single morpheme. Take den blauen Regenschirm: The article ending -en carries the information masculine, singular, accusative while the adjective ending -en carries the information case/numerus not nominative singular, case/numerus/genus combination not feminine singular or neuter singular and accusative. There is much more information condensed in the article than in the adjective ending.
Fusional languages tend to avoid double marking somewhat. It is enough if all three bits of information — case, numerus and genus — are encoded into the combination of all possible markers — article ending and potential adjective endings. And even then they tend to live with a certain grammatical or morphological ambiguity — note the identical forms of accusative and nominative if the noun is not masculine. Therefore, it is highly unlikely for a fusional language to display a pattern as you noted.
Agglutinative languages such as Finnish tend to behave differently if they display congruence, since every bit of information is coded into its very own morpheme. Thus it makes sense for sinisen sateenvarjon (accusative singular), siniset sateenvarjot (accusative or nominative plural), sinistä sateenvarjoa (partitive singular) or sinisiä sateenvarjoja (partitive plural) to have repeated endings.*
So the short, but painful answer to your question is:
Because that is how German evolved.
*: If this looks slightly confusing, note that ä and a in endings are equivalent and depend only on the vowels present in the stem, that i turns into j between two vowels and e is sometimes dropped.
Nach der/die/das hat das Adjektiv die Endung e, alle übrigen Endungen sind -en.
