Why do they swear in "Ashmas Shomron"?
Radak: Now they despise Devar Hashem, and swear in Ashmas Shomron, i.e. the calf in Beis Kel. It is attributed to Shomron, which was the capitol of Malchus Efrayim, for they supported Avodah of the calves and the nation followed them.
Malbim: These are the calves that the kings in Shomron made.
Why do they say "Chei Elohecha Dan?
Rashi: One of the calves that Yaravam set up was in Dan.
Radak: When one swears to his friend, he says 'Chei Elohecha Asher b'Dan', i.e. the calf.
Why do they say "v'Chei Derech Be'er Sheva"?
Radak: From Be'er-Sheva, at the southern end of the Techum (border of Yisrael), they went to the calf in Dan, at the other (northern) end, like it says "mi'Dan v'Ad Be'er-Shava" (Melachim I, 5:5). They clung so much to idolatry, that they thought that they will receive great reward if they go to serve the calf there - "va'Yelechu ha'Am Lifnei ha'Echad Ad Dan" (ibid. 12:30). They swore 'in the life of the one that we went to serve it via the road of Be'er-Sheva.'
Radak citing Targum Yonasan: Derech is [the way of living], like "v'Hodata Lahem Es ha'Derech Yelechu Vah" (Shemos 18:20).
Malbim: This is the calf [that one reaches it] via the road of Be'er-Sheva; from there, they went to Beis Kel.
What is the significance of "v'Nafalu v'Lo Yakumu Od"?
Radak: Those who swear and say so, will fall and not rise.
Malbim: This is like it said above "Naflah Lo Sosif Kum" (5:2). Hashem said "Dirshuni vi'Chyu; v'Al Tidreshu Beis Kel... u'Ve'er Sheva Lo Sa'avoru" (ibid. 3).
