If you really feel a need to digress into some history, knock yourself out...
The instability of the system was, as already noted, aptly reflected in the numerous changes of government. No Prime Minister spent much longer than a year in office before being removed, usually by the collapse of his coalition, after which he would feel compelled to resign. The governments were seen as impotent, unable to exert authority over any section of the population either at home or abroad. There were splits over the question of the European Defense Community with neither a definitive "yes" or "no" being offered as a response. The attempts to retain France's interests in South East Asia were ultimately unsuccessful, efforts to collect taxes were thwarted by the Poujadists, and inflation and agricultural overproduction (reflecting the still strong influence of farmers) were constant problems. The cause of this impotence was an inability of cabinets to govern effectively; their duration was short, and members were often more interested in securing their own political future knowing that in all likelihood the cabinet would not last for long than doing what was expeditious for the nation. Prime Ministers, knowing that they did not have an assured majority in the National Assembly, often extended their term in office by doing nothing whatsoever, which only exacerbated the inaction. This inability to take decisions was known as immobilisme. Explanations that have been offered for the existence of immobilisme include the irreconcilable ideologies of the various parties, a deeply divided society that was merely reflected (and possibly amplified) in the party system, and as I have already mentioned, deputés who were more concerned about their political future than the interests given to principles or policy.
The nature of the party system made coalition necessary in order to guarantee a majority in the National Assembly. However, these were fragile affairs, and often self-destructed with consummate ease. There were around eight political parties represented in the Assembly at any one time, with a minimum of three needed to gain the requisite majority. However, the large share of votes on extreme right (Gaullists and Poujadists) and left (Communists) made these votes inaccessible to any moderate party of the center, as they would alienate both left and right, and be facing a defeat. This contributed in part to the immobilisme. Secondly, even within a coalition, opinion was so sharply divided as to render any form of radical legislation almost impossible to gain support from within the cabinet and coalition, hence the lack of government activity at this time.
There has always been deep social diversity and division in France reflected in disputes between working-class and bourgeoisie, urban and rural areas, etc. French people have never seen themselves as a single society, but the prevalent view was of an atomist society of individuals. The saliency of these divisions has often manifested itself in bloody revolutions, and indeed some commentators have spoken of how close France came to open civil war over Algeria. However, for the majority of the time of the Fourth Republic, these divisions were represented by the fragmented party system. There were also what are known as "surge" parties, a newly founded party or movement that gain support rapidly and cuts across the social divisions of the stable parties, which added an extra measure of instability, especially given that their popularity was often very short-lived. This entire system had a large responsibility for the failure of the Fourth Republic.
The final factor of those three already mentioned which contributed to immobilisme was the conflict between the deputés and party activists; the former tended to support one another and look after their careers, whilst the latter were dissatisfied with the compromises made by the deputés, even when such compromises were necessary. This dissatisfaction was often manifested through a waning of support for the cabinet from party activists, which sometimes prompted it to resign. In this way, grassroots often had a devastating effect on the government of the Fourth Republic.
As well as this, there was also the contributing factor of massive instability in the cabinet. Cabinets were invested by a majority of the National Assembly, and remained as long as they believed they could assure a majority to pass pieces of legislation. The trouble was that, as already pointed out above, the extremes of the political spectrum, which made up about one-third of the votes available in the Assembly, couldn't be relied on. Thus, when splits occurred in the coalition partners, there was often a rapid erosion of the majority, followed by the resignation of the Prime Minister when it became narrow. Often the Cabinet was faced with a choice between immobility, doing nothing and remaining in power, or attempting to press on with government business, which often led to rapid overthrow. This was a major criticism of the Fourth Republic, noted especially by Debré, who traced the lack of government authority down to the way in which a cabinet could be defeated, the possibility of any individual member of the cabinet advancing his career from its break-up, and the lack of responsibility on the part of the deputés for their voting patterns in the Assembly. Ultimately, thought, it was the fact that cabinets were loose, complex, and temporary coalitions rather than agreements to govern that caused this instability; the ideological incompatibility of the factions within the parties within the cabinet and the conflicting ambitions of the potential government leaders within the cabinet a result of this. Thus immobilisme was a combination of cabinet weakness and instability.
The crisis in Algeria was the catalyst that brought about the downfall of the Fourth Republic. As already mentioned, France had been relinquishing her colonies after the Second World War, most notably in South East Asia, but also in Africa. However, Algeria was not to be allowed the same freedom. Jacques Soustelle, governor-general of Algeria from 1955 declared "It is precisely because we have lost Indo-China, Tunisia, and Morocco that we must not, at any price, in any way, and under any pretext, lose Algeria." He was speaking nearly three months after the Front de Libération Nationale (FLN), a grouping of nationalist Algerian terrorists launched an armed insurrection into the South of the country in an attempt to drive the French out and achieve a sovereign Algerian state. Despite the fact that fighting between government troops and the rebels continued for eight years, the French never acknowledged that there was a war in Algeria, only internal problems of public order, or "events." Censorship was strictly imposed, which hid from view the fact that the French army was engaged in brutal and savage repression and torture. Under dictatorial powers granted to the minister-resident in Algiers in 1956, and then passed on to the head of the French army in Algeria, a campaign of terror was instituted throughout the country to flush out the rebels. However, it became apparent that they had the popular support of the people of Algeria, which lead to the use of torture, and a backlash back home when it became known a year later. Much like in Northern Ireland, there were two sides to the population; there were one million French nationals living in Algeria, who wished to remain a part of France. These "pieds-noirs" (so called because of the black shoes they wore which distinguished them from the native population who went barefoot), fearing that they would eventually lose, took matters into their own hands, and aligned themselves with right-wing organizations and disaffected army officers to organize a mass strike on 13 May 1958, and then invade the governor-general's office and create a committee of public safety under the head of the French army. They called in De Gaulle to form a government of public safety to protect French Algeria. De Gaulle distanced himself from the coup, but said he was ready to assume powers of government. The coup initially strengthened the Paris government's hand, and the new Prime Minister Pflimlin who was being invested at the time. Unfortunately, faced with the prospect of the army in Algeria mounting a coup in France, or Communists protesting against what they saw as De Gaulle becoming another dictator in the mould of Pétain, and threatening an insurrection, Pflimlin eventually resigned, and President Coty asked De Gaulle to form a new government. De Gaulle asked for time to restore order, draft a new constitution, and put it to a referendum. In a state of panic, the Assembly split and voted him these powers. Thus ended the Fourth Republic.
How far Algeria contributed to the downfall of the Fourth Republic has been a matter for enormous debate. Had there been no crisis, would the institutions of government have survived and become more stable? This seems unlikely; the short-lived nature of governments was still present - there were no indications of any lengthening in government's tenure; neither was there any sign of growing stability within Cabinet. In short, all the deficiencies that have been described above would have eventually needed to be overhauled to make way for a more effective government; Algeria provided the catalyst for such a change. Indeed, it is perhaps a testament to De Gaulle that he was able to successfully solve the Algerian crisis as well as overhaul the political system in France. This was no easy feat, given the severity of each of the problems, and the need to restore confidence among the French people. It has been suggested that only a man of De Gaulle's stature could have attempted, let alone seen a successful resolution, to both conflicts.
In conclusion, the Fourth Republic contained many structural weaknesses that were not conducive to any long-term stability and effectiveness in government. Algeria was the catalyst that prompted the change that De Gaulle was able to provide. Without the crisis, it is difficult to say what the results would have been for the Fourth Republic, but it would surely not have survived for much longer. In terms of importance, the inherent problems of the system were more important in its downfall than Algeria, which merely accelerated the process.