The answer to your question lies in the Algerian War and the crisis of 1958. I will briefly touch on 1961 at the end, but the important event really took place 3 years previously.

The Algiers Crisis is a prime example of both the complexity and the role of the French army in Algeria. What I am about to write is at best a summary of a very complex political crisis. I would recommend that anyone interested in this to read one of the excellent books on the Algerian War available in English.

The French Army

By 1958, there was a very present fear among the leadership of the French army in Algeria as well as their fellow officers. It was a painful reminder of what the French army had experienced in Indochina just 4 years ago in which, in their own words, they had been abandoned by the politicians back home. Essentially, a French stab-in-the-back myth. When the French army properly entered the Algerian War, it did so with a vengeance. It would not let Algeria, a place that was considered to be as French as Dunkirk itself, be abandoned because of the politicians back in metropolitan France. They were not going to allow their comrades to have died in vain and they would do anything to keep Algerian French. When there were talks of the current government, which hadn't even been properly been formed due to previous events, began speaking of possible negotiations with the FLN, the army had enough.

The French settlers

If you look at the history of the French settlers in Algeria in the 20th century, you will perhaps characterize them as being arrogant and pushy in the political realm. They wanted nothing, nothing at all, to ruin the hegemony they had in Algeria, a sensitive balance in which the French settlers were at the top and the Muslim was at the bottom. They had prevented any major pre-war reform that would have given the Muslim population a slight push forward and had objected to any and all talk of giving equal rights to Muslims - even when done through democratic means. For example, in the 1947 municipal election, there was a very clear case of rigging in favor of the settlers and preventing a perfectly legal democratic party to enter their assembly. With that mindset, the most extreme of the settlers looked towards the army for salvation in 1958 when there was a fear that the current government would abandon them.

It was a powder keg that was ready to blow. A conspiracy was afoot and it was in May 1958 that it would all come together.

One of the most telling signs of impending army involvement in national politics came in the form of a telegram sent by General Salan to General Ely in France:

The present crisis shows that the political parties are profoundly divided over the Algerian question. The Press permits one to think that the abandonment of Algeria would be envisaged in the diplomatic processes which would begin with negotiations aiming at a "cease-fire" . . . .

The army in Algeria is troubled by recognition of its responsibility towards the men who are fighting and risking a useless sacrifice if the representatives of the nation are not determined to maintain Algérie francaise . . . .

The French army, in its unanimity, would feel outraged by the abandonment of this national patrimony. One cannot predict how it would react in its despair . . . .

I request you to bring to the attention of the President of the Republic our anguish, which only a government firmly determined to maintain our flag in Algeria can efface.

It all came together on May 13, 1958. What had begun as a ceremony to pay homage to three French soldiers that had been executed by the FNL later turned into a demonstration or rather, mass meeting of French settlers by the monument aux morts. The hardline settler "leadership" had whipped them up to a frenzy and when General Salan arrived at the scene, shouts of "put the army in power!" were common and when the ceremony was over, there was little to stop the settlers from taking over the French government in Algeria. Things happened quickly from here. The army was thrust into power and adopted it willingly. The leadership demanded Charles de Gaulle to take power in France (who was expected to be the sort of person to not want to give up Algeria). When this failed to materialize quickly, plans were drawn up and threats were given to France that if things didn't start moving faster, they'd take care of it themselves (i.e. a military intervention in France). This was not empty words however and plans were drawn up for it. Corsica was "invaded" on May 24th and there was a very present fear in Paris of a intervention by French paras. De Gaulle finally agreed to form a government on May 30th.

With Charles de Gaulle in power, he essentially demanded dictatorial powers for a limited time and then went on to put forward a new constitution. This constitution was then put down to the ballots and it led an almost overwhelming 'yes' in favor of a new constitution. 79.25 % voted yes. The Fourth Republic had died in Algeria. The Fifth Republic had come out of its ashes and Charles de Gaulle took the stage on December 21st 1958 as its first president.

The hope was that De Gaulle wouldn't betray Algeria, but they were sorely mistaken. A referendum was held in 1961, with French voters overwhelmingly voting for granting Algerian independence. A few months later, conservative elements of the French Army in Algeria rose up in an attempted coup - the Generals Putsch - which quickly failed, and resulted in the disbandment of part of the Foreign Legion.