Why did the United States decide to cross over the 38th parallel to North Korea after they attacked South Korea?

The question asks about General MacArthur's decision to cross the 38th parallel thus 'invading North Korea'. The answer to that question is that General MacArthur, heading a U.N force tasked with repelling North Korean aggression, saw that South Korean people were deeply divided and the US supported regime was corrupt, contained collaborators, and was militarily less capable in every way than the North. Thus, the North had to be fought and beaten on its own turf. Failure to do so would repeat the mistakes made by the US in their support for the corrupt KMT in China. What MacArthur did not know, or admit, was that the Communist Chinese had a capable army and that they were prepared to pay any price to push back the U.N forces to the 38th parallel. Actually, Truman had only authorised MacArthur to go north if the Chinese or Soviets didn't have boots on the ground there. MacArthur underestimated the enemy as well as his own prestige and thus himself became a casualty of that bloody and futile war.

To be clear- the US invaded North Korea because

1) it thought China and the Soviets would not put 'boots on the ground'

2) it had a UN mandate to do so. Moreover, victory in Korea would go a long way in confirming the legitimacy of the Chiang Kai Shek regime in Taiwan's occupying a Permanent Seat, with veto power, in the UN Security Council

3) Syngman Rhee's South Korean regime was corrupt and full of collaborators. It was authoritarian and deeply unpopular. This is the true reason why the North invaded. Rhee seemed to be bumping off other Nationalist leaders who would have prepared to work with the North for peaceful re-unification.

Prior to General MacArthur's decision to invade North Korea, it should be remembered that the U.S felt it had just lost China to the Communists. The USSR was boycotting the UN in an effort to get the US to recognise Mao's regime and reject the regime which Chiang Kai Shek had established in Taiwan after he and his Kuomintang party (KMT) fled before the Mao's Army.

The North Korean invasion of the South was interpreted by the Americans as an attempt to force the recognition of Communist China- which would then take Taiwan's seat as a Permanent Member of the Security Council- a seat it retained till 1971 and Nixon's rapproachement with Mao.

Furthermore, if the Americans did not repel this aggression, then the Soviets might make a move against West Germany or else Greece, where there was a strong Leftist grass-roots movement.

Since the Soviets had conducted a successful atom bomb test the previous year, the US would have to 'put boots on the ground'. It could not rely upon the threat of another Hiroshima. The credibility of the 'Truman Doctrine'- i.e. the promise to assist 'free people' from Communism- had to be very firmly demonstrated and established.

Furthermore, the Soviet absence from the Security Council meant that they couldn't veto the military campaign receiving the imprimatur of the U.N General Assembly. This would be an important moral victory.

By 1950, left-leaning 'New Dealers' had been purged. Anti-Communist sentiment was on the rise. The loss of China was blamed on 'crypto-communists' who certainly had helped the Soviets to get the atom bomb. Another factor was that there was a fear of a recurrence of Economic Depression. Thus a 'War boom' was anticipated by the markets.

It should be mentioned that Syngman Rhee, the South Korean leader, had developed close ties with the American Ivy League elite as well as with Christian leaders. On the other hand, his prickly character and authoritarian tendencies caused the State Department to recoil from him. However the State Department was itself considered 'soft' on Communism and thus their dislike of Rhee- who, from the Leftist point of view, had precipitated the crisis- was no impediment. American Generals liked Rhee and his strong anti-communism impressed Truman who announced the Truman Doctrine after a visit to Korea. It may be that if Rhee had not taken such a hard line and also if he did not appear to be guilty of the assassination of an opponent, then the North might not have launched an invasion hoping to be welcomed by ordinary Koreans disgusted with the corruption of the elites, who had previously served the Japanese.

In other words, the US decision in favour of military intervention arose out of the fact that it had given more weighting to its Generals than its diplomats in that theater of operations. This was itself dictated by the perception that the State Department had 'lost China'. Korea was considered the most suitable front to deter Communist aggression which might otherwise threaten Central and South Eastern Europe- thus reigniting a global conflagration.

This is a very basic and straight-forward question which can be answered by looking at the first few paragraphs on wikipedia. Since I'm not sure of the specifics the basic outline I've written below.

[It] was a product of the Cold War between United States and the Soviet Union. Korea had been split into two sovereign nations. Both governments claimed to be sole legitimate government of the whole of Korea and neither accepted the border as permanent.

The conflict escalated into open warfare when North Korean forces - supported by the Soviet Union and China - moved into the South ... The UN Security Council authorised the formation and dispatch of UN forces into Korea to repel what was recognised as a North Korean invasion.

Twenty-one countries contributed personnel with the United States providing 90%

Essentially, Korea was under Japanese hegemony after Japan militarised in its attempt to modernise in imitation of the West, and in further imitation of Europe it looked for colonies abroad - in China and Korea. With collapse of Japanese power in the aftermath of the nuclear bombing of Nagasaki and Hiroshima by the United States military it's sphere of influence was split between the Soviet Union and The United States.

In Korea, this is the origin of the 38th Parallel as a demarcation and boundary between North and South Korea. This was drawn up by US colonels Dean Rusk and Charlea Bonesteel III and incorporated into US General order 1.

They doubted whether the Soviet Union would cooperate. However the Red Army fully complied with this decision under Stalins policy of wartime cooperation and the Red Army reaching this parallel first waited there for three weeks to await the arrival of US forces from the South.